Functions that compute some of the sequences in Sloane’s tables

EXAMPLES:

Type sloane.[tab] to see a list of the sequences that are defined.

Sage

sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a(1)
1
sage: a(6)
4
sage: a(100)
9

Python

>>> from sage.all import *
>>> a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
>>> a(Integer(1))
1
>>> a(Integer(6))
4
>>> a(Integer(100))
9

Type d._eval?? to see how the function that computes an individual term of the sequence is implemented.

The input must be a positive integer:

Sage

sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

You can also change how a sequence prints:

Sage

sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a.rename('(..., tau(n), ...)')
sage: a
(..., tau(n), ...)
sage: a.reset_name()
sage: a
The integer sequence tau(n), which is the number of divisors of n.

Python

>>> from sage.all import *
>>> a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
>>> a.rename('(..., tau(n), ...)')
>>> a
(..., tau(n), ...)
>>> a.reset_name()
>>> a
The integer sequence tau(n), which is the number of divisors of n.

See also

• If you want to get more informations relative to a sequence (references, links, examples, programs, …), you can use the On-Line Encyclopedia of Integer Sequences provided by the OEIS module.

• If you plan to do a lot of automatic searches for subsequences, you should consider installing SloaneEncyclopedia, a local partial copy of the OEIS.

AUTHORS:

• William Stein: framework

• Jaap Spies: most sequences

• Nick Alexander: updated framework

class sage.combinat.sloane_functions.A000001

Bases: SloaneSequence

Number of groups of order \(n\).

INPUT:

• n – positive integer

OUTPUT: integer

EXAMPLES:

Sage

sage: a = sloane.A000001;a
Number of groups of order n.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(9)
2
sage: a.list(16)
[1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
sage: a(60)
13

Python

>>> from sage.all import *
>>> a = sloane.A000001;a
Number of groups of order n.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(9))
2
>>> a.list(Integer(16))
[1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
>>> a(Integer(60))
13

AUTHORS:

• Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A000004

Bases: SloaneSequence

The zero sequence.

INPUT:

• n – non negative integer

EXAMPLES:

Sage

sage: a = sloane.A000004; a
The zero sequence.
sage: a(1)
0
sage: a(2007)
0
sage: a.list(12)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> a = sloane.A000004; a
The zero sequence.
>>> a(Integer(1))
0
>>> a(Integer(2007))
0
>>> a.list(Integer(12))
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

AUTHORS:

• Jaap Spies (2006-12-10)

class sage.combinat.sloane_functions.A000005

Bases: SloaneSequence

The sequence \(tau(n)\), which is the number of divisors of \(n\).

This sequence is also denoted \(d(n)\) (also called \(\tau(n)\) or \(\sigma_0(n)\)), the number of divisors of \(n\).

INPUT:

• n – positive integer

EXAMPLES:

Sage

sage: d = sloane.A000005; d
The integer sequence tau(n), which is the number of divisors of n.
sage: d(1)
1
sage: d(6)
4
sage: d(51)
4
sage: d(100)
9
sage: d(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: d.list(10)
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

Python

>>> from sage.all import *
>>> d = sloane.A000005; d
The integer sequence tau(n), which is the number of divisors of n.
>>> d(Integer(1))
1
>>> d(Integer(6))
4
>>> d(Integer(51))
4
>>> d(Integer(100))
9
>>> d(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> d.list(Integer(10))
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

AUTHORS:

• Jaap Spies (2006-12-10)

• William Stein (2007-01-08)

class sage.combinat.sloane_functions.A000007

Bases: SloaneSequence

The characteristic function of 0: \(a(n) = 0^n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000007;a
The characteristic function of 0: a(n) = 0^n.
sage: a(0)
1
sage: a(2)
0
sage: a(12)
0
sage: a.list(12)
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> a = sloane.A000007;a
The characteristic function of 0: a(n) = 0^n.
>>> a(Integer(0))
1
>>> a(Integer(2))
0
>>> a(Integer(12))
0
>>> a.list(Integer(12))
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

AUTHORS:

• Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000008

Bases: SloaneSequence

Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000008;a
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
16
sage: a.list(14)
[1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]

Python

>>> from sage.all import *
>>> a = sloane.A000008;a
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(13))
16
>>> a.list(Integer(14))
[1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]

AUTHOR:

• 10. Gaski (2009-05-29)

class sage.combinat.sloane_functions.A000009

Bases: SloaneSequence

Number of partitions of \(n\) into odd parts.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000009;a
Number of partitions of n into odd parts.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
18
sage: a.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

Python

>>> from sage.all import *
>>> a = sloane.A000009;a
Number of partitions of n into odd parts.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(13))
18
>>> a.list(Integer(14))
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

AUTHOR:

• Jaap Spies (2007-01-30)

cf()

EXAMPLES:

Sage

sage: it = sloane.A000009.cf()
sage: [next(it) for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

Python

>>> from sage.all import *
>>> it = sloane.A000009.cf()
>>> [next(it) for i in range(Integer(14))]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

list(n)

EXAMPLES:

Sage

sage: sloane.A000009.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

Python

>>> from sage.all import *
>>> sloane.A000009.list(Integer(14))
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

class sage.combinat.sloane_functions.A000010

Bases: SloaneSequence

The integer sequence A000010 is Euler’s totient function.

Number of positive integers \(i < n\) that are relative prime to \(n\). Number of totatives of \(n\).

Euler totient function \(\phi(n)\): count numbers \(n\) and prime to \(n\). euler_phi is a standard Sage function implemented in PARI

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000010; a
Euler's totient function
sage: a(1)
1
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(11)
10
sage: a.list(12)
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a = sloane.A000010; a
Euler's totient function
>>> a(Integer(1))
1
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(11))
10
>>> a.list(Integer(12))
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHORS:

• Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000012

Bases: SloaneSequence

The all 1’s sequence.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000012; a
The all 1's sequence.
sage: a(1)
1
sage: a(2007)
1
sage: a.list(12)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Python

>>> from sage.all import *
>>> a = sloane.A000012; a
The all 1's sequence.
>>> a(Integer(1))
1
>>> a(Integer(2007))
1
>>> a.list(Integer(12))
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

AUTHORS:

• Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000015

Bases: SloaneSequence

Smallest prime power \(\geq n\) (where \(1\) is considered a prime power).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000015; a
Smallest prime power >= n.
sage: a(1)
1
sage: a(8)
8
sage: a(305)
307
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A000015; a
Smallest prime power >= n.
>>> a(Integer(1))
1
>>> a(Integer(8))
8
>>> a(Integer(305))
307
>>> a(-Integer(4))
Traceback (most recent call last):
...
ValueError: input n (=-4) must be a positive integer
>>> a.list(Integer(12))
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000016

Bases: SloaneSequence

Sloane’s A000016

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000016; a
Sloane's A000016.
sage: a(1)
1
sage: a(0)
1
sage: a(8)
16
sage: a(75)
251859545753048193000
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

Python

>>> from sage.all import *
>>> a = sloane.A000016; a
Sloane's A000016.
>>> a(Integer(1))
1
>>> a(Integer(0))
1
>>> a(Integer(8))
16
>>> a(Integer(75))
251859545753048193000
>>> a(-Integer(4))
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
>>> a.list(Integer(12))
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000027

Bases: SloaneSequence

The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

The following examples are tests of SloaneSequence more than A000027.

EXAMPLES:

Sage

sage: s = sloane.A000027; s
The natural numbers.
sage: s(10)
10

Python

>>> from sage.all import *
>>> s = sloane.A000027; s
The natural numbers.
>>> s(Integer(10))
10

Index n is interpreted as _eval(n):

Sage

sage: s[10]
10

Python

>>> from sage.all import *
>>> s[Integer(10)]
10

Slices are interpreted with absolute offsets, so the following returns the terms of the sequence up to but not including the third term:

Sage

sage: s[:3]
[1, 2]
sage: s[3:6]
[3, 4, 5]
sage: s.list(5)
[1, 2, 3, 4, 5]

Python

>>> from sage.all import *
>>> s[:Integer(3)]
[1, 2]
>>> s[Integer(3):Integer(6)]
[3, 4, 5]
>>> s.list(Integer(5))
[1, 2, 3, 4, 5]

link = 'http://oeis.org/classic/A000027'

class sage.combinat.sloane_functions.A000030

Bases: SloaneSequence

Initial digit of \(n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000030; a
Initial digit of n
sage: a(0)
0
sage: a(1)
1
sage: a(8)
8
sage: a(454)
4
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

Python

>>> from sage.all import *
>>> a = sloane.A000030; a
Initial digit of n
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(8))
8
>>> a(Integer(454))
4
>>> a(-Integer(4))
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
>>> a.list(Integer(12))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000032

Bases: SloaneSequence

Lucas numbers (beginning at 2): \(L(n) = L(n-1) + L(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000032; a
Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
sage: a(0)
2
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

Python

>>> from sage.all import *
>>> a = sloane.A000032; a
Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
>>> a(Integer(0))
2
>>> a(Integer(1))
1
>>> a(Integer(8))
47
>>> a(Integer(200))
627376215338105766356982006981782561278127
>>> a(-Integer(4))
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
>>> a.list(Integer(12))
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000035

Bases: SloaneSequence

A simple periodic sequence.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000035;a
A simple periodic sequence.
sage: a(0.0)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
sage: a(1)
1
sage: a(2)
0
sage: a(9)
1
sage: a.list(10)
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

Python

>>> from sage.all import *
>>> a = sloane.A000035;a
A simple periodic sequence.
>>> a(RealNumber('0.0'))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
>>> a(Integer(1))
1
>>> a(Integer(2))
0
>>> a(Integer(9))
1
>>> a.list(Integer(10))
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

AUTHORS:

• Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A000040

Bases: SloaneSequence

The prime numbers.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000040; a
The prime numbers.
sage: a(1)
2
sage: a(8)
19
sage: a(305)
2011
sage: a.list(12)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A000040; a
The prime numbers.
>>> a(Integer(1))
2
>>> a(Integer(8))
19
>>> a(Integer(305))
2011
>>> a.list(Integer(12))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-17)

class sage.combinat.sloane_functions.A000041

Bases: SloaneSequence

\(a(n)\) = number of partitions of \(n\) (the partition numbers).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000041;a
a(n) = number of partitions of n (the partition numbers).
sage: a(0)
1
sage: a(2)
2
sage: a(8)
22
sage: a(200)
3972999029388
sage: a.list(9)
[1, 1, 2, 3, 5, 7, 11, 15, 22]

Python

>>> from sage.all import *
>>> a = sloane.A000041;a
a(n) = number of partitions of n (the partition numbers).
>>> a(Integer(0))
1
>>> a(Integer(2))
2
>>> a(Integer(8))
22
>>> a(Integer(200))
3972999029388
>>> a.list(Integer(9))
[1, 1, 2, 3, 5, 7, 11, 15, 22]

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000043

Bases: SloaneSequence

Primes \(p\) such that \(2^p - 1\) is prime. \(2^p - 1\) is then called a Mersenne prime.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000043;a
Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
sage: a(1)
2
sage: a(2)
3
sage: a(39)
13466917
sage: a(40)
Traceback (most recent call last):
...
IndexError: list index out of range
sage: a.list(12)
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

Python

>>> from sage.all import *
>>> a = sloane.A000043;a
Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
>>> a(Integer(1))
2
>>> a(Integer(2))
3
>>> a(Integer(39))
13466917
>>> a(Integer(40))
Traceback (most recent call last):
...
IndexError: list index out of range
>>> a.list(Integer(12))
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000045

Bases: SloaneSequence

Sequence of Fibonacci numbers, offset 0,4.

REFERENCES:

• S. Plouffe, Project Gutenberg, The First 1001 Fibonacci Numbers, http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt

We have one more. Our first Fibonacci number is 0.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000045; a
Fibonacci numbers with index n >= 0
sage: a(0)
0
sage: a(1)
1
sage: a.list(12)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a = sloane.A000045; a
Fibonacci numbers with index n >= 0
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a.list(Integer(12))
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHORS:

• Jaap Spies (2007-01-13)

fib()

Returns a generator over all Fibonacci numbers, starting with 0.

EXAMPLES:

Sage

sage: it = sloane.A000045.fib()
sage: [next(it) for i in range(10)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Python

>>> from sage.all import *
>>> it = sloane.A000045.fib()
>>> [next(it) for i in range(Integer(10))]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

list(n)

EXAMPLES:

Sage

sage: sloane.A000045.list(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Python

>>> from sage.all import *
>>> sloane.A000045.list(Integer(10))
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

class sage.combinat.sloane_functions.A000069

Bases: SloaneSequence

Odious numbers: odd number of 1’s in binary expansion.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000069; a
Odious numbers: odd number of 1's in binary expansion.
sage: a(0)
1
sage: a(2)
4
sage: a.list(9)
[1, 2, 4, 7, 8, 11, 13, 14, 16]

Python

>>> from sage.all import *
>>> a = sloane.A000069; a
Odious numbers: odd number of 1's in binary expansion.
>>> a(Integer(0))
1
>>> a(Integer(2))
4
>>> a.list(Integer(9))
[1, 2, 4, 7, 8, 11, 13, 14, 16]

AUTHORS:

• Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A000073

Bases: SloaneSequence

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, …

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000073;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(11)
149
sage: a.list(12)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

Python

>>> from sage.all import *
>>> a = sloane.A000073;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
>>> a(Integer(0))
0
>>> a(Integer(1))
0
>>> a(Integer(2))
1
>>> a(Integer(11))
149
>>> a.list(Integer(12))
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

AUTHORS:

• Jaap Spies (2007-01-19)

list(n)

EXAMPLES:

Sage

sage: sloane.A000073.list(10)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

Python

>>> from sage.all import *
>>> sloane.A000073.list(Integer(10))
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

class sage.combinat.sloane_functions.A000079

Bases: SloaneSequence

Powers of 2: \(a(n) = 2^n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000079;a
Powers of 2: a(n) = 2^n.
sage: a(0)
1
sage: a(2)
4
sage: a(8)
256
sage: a(100)
1267650600228229401496703205376
sage: a.list(9)
[1, 2, 4, 8, 16, 32, 64, 128, 256]

Python

>>> from sage.all import *
>>> a = sloane.A000079;a
Powers of 2: a(n) = 2^n.
>>> a(Integer(0))
1
>>> a(Integer(2))
4
>>> a(Integer(8))
256
>>> a(Integer(100))
1267650600228229401496703205376
>>> a.list(Integer(9))
[1, 2, 4, 8, 16, 32, 64, 128, 256]

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000085

Bases: SloaneSequence

Number of self-inverse permutations on \(n\) letters, also known as involutions; number of Young tableaux with \(n\) cells.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000085;a
Number of self-inverse permutations on n letters.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
2
sage: a(12)
140152
sage: a.list(13)
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

Python

>>> from sage.all import *
>>> a = sloane.A000085;a
Number of self-inverse permutations on n letters.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(12))
140152
>>> a.list(Integer(13))
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

AUTHORS:

• Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A000100

Bases: SloaneSequence

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000100;a
Number of compositions of n in which the maximum part size is 3.
sage: a(0)
0
sage: a(1)
0
sage: a(2)
0
sage: a(3)
1
sage: a(11)
360
sage: a.list(12)
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

Python

>>> from sage.all import *
>>> a = sloane.A000100;a
Number of compositions of n in which the maximum part size is 3.
>>> a(Integer(0))
0
>>> a(Integer(1))
0
>>> a(Integer(2))
0
>>> a(Integer(3))
1
>>> a(Integer(11))
360
>>> a.list(Integer(12))
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000108

Bases: SloaneSequence

Catalan numbers: \(C_n = \frac{\binom{2n}{n}}{n+1} = \frac{(2n)!}{n!(n+1)!}\).

Also called Segner numbers.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000108;a
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
sage: a(0)
1
sage: a.offset
0
sage: a(8)
1430
sage: a(40)
2622127042276492108820
sage: a.list(9)
[1, 1, 2, 5, 14, 42, 132, 429, 1430]

Python

>>> from sage.all import *
>>> a = sloane.A000108;a
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
>>> a(Integer(0))
1
>>> a.offset
0
>>> a(Integer(8))
1430
>>> a(Integer(40))
2622127042276492108820
>>> a.list(Integer(9))
[1, 1, 2, 5, 14, 42, 132, 429, 1430]

AUTHORS:

• Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000110

Bases: ExponentialNumbers

The sequence of Bell numbers.

The Bell number \(B_n\) counts the number of ways to put \(n\) distinguishable things into indistinguishable boxes such that no box is empty.

Let \(S(n, k)\) denote the Stirling number of the second kind. Then

\[B_n = \sum{k=0}^{n} S(n, k).\]

INPUT:

• n – non negative integer

OUTPUT:

• integer – \(B_n\)

EXAMPLES:

Sage

sage: a = sloane.A000110; a
Sequence of Bell numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
sage: a.list(10)
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

Python

>>> from sage.all import *
>>> a = sloane.A000110; a
Sequence of Bell numbers
>>> a.offset
0
>>> a(Integer(0))
1
>>> a(Integer(100))
47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
>>> a.list(Integer(10))
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

AUTHORS:

• Nick Alexander

class sage.combinat.sloane_functions.A000120

Bases: SloaneSequence

1’s-counting sequence: number of 1’s in binary expansion of \(n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000120;a
1's-counting sequence: number of 1's in binary expansion of n.
sage: a(0)
0
sage: a(2)
1
sage: a(12)
2
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

Python

>>> from sage.all import *
>>> a = sloane.A000120;a
1's-counting sequence: number of 1's in binary expansion of n.
>>> a(Integer(0))
0
>>> a(Integer(2))
1
>>> a(Integer(12))
2
>>> a.list(Integer(12))
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

AUTHORS:

• Jaap Spies (2007-01-26)

f(n)

EXAMPLES:

Sage

sage: [sloane.A000120.f(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

Python

>>> from sage.all import *
>>> [sloane.A000120.f(n) for n in range(Integer(10))]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

class sage.combinat.sloane_functions.A000124

Bases: SloaneSequence

Central polygonal numbers (the Lazy Caterer’s sequence): \(n(n+1)/2 + 1\).

Or, maximal number of pieces formed when slicing a pancake with \(n\) cuts.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000124;a
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
sage: a(0)
1
sage: a(1)
2
sage: a(2)
4
sage: a(9)
46
sage: a.list(10)
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

Python

>>> from sage.all import *
>>> a = sloane.A000124;a
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
>>> a(Integer(0))
1
>>> a(Integer(1))
2
>>> a(Integer(2))
4
>>> a(Integer(9))
46
>>> a.list(Integer(10))
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000129

Bases: RecurrenceSequence2

Pell numbers: \(a(0) = 0\), \(a(1) = 1\); for \(n > 1\), \(a(n) = 2a(n-1) + a(n-2)\).

Denominators of continued fraction convergents to \(\sqrt 2\).

See also A001333

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000129;a
Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
sage: a(0)
0
sage: a(2)
2
sage: a(12)
13860
sage: a.list(12)
[0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

Python

>>> from sage.all import *
>>> a = sloane.A000129;a
Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(2))
2
>>> a(Integer(12))
13860
>>> a.list(Integer(12))
[0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000142

Bases: SloaneSequence

Factorial numbers: \(n! = 1 \cdot 2 \cdot 3 \cdots n\)

Order of symmetric group \(S_n\), number of permutations of \(n\) letters.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000142;a
Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
sage: a(0)
1
sage: a(8)
40320
sage: a(40)
815915283247897734345611269596115894272000000000
sage: a.list(9)
[1, 1, 2, 6, 24, 120, 720, 5040, 40320]

Python

>>> from sage.all import *
>>> a = sloane.A000142;a
Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
>>> a(Integer(0))
1
>>> a(Integer(8))
40320
>>> a(Integer(40))
815915283247897734345611269596115894272000000000
>>> a.list(Integer(9))
[1, 1, 2, 6, 24, 120, 720, 5040, 40320]

AUTHORS:

• Jaap Spies (2007-01-12)

class sage.combinat.sloane_functions.A000153

Bases: ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-2)*a(n-2)\), with \(a(0) = 0\), \(a(1) = 1\).

With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000153; a
a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
sage: a(0)
0
sage: a(1)
1
sage: a(8)
82508
sage: a(20)
10315043624498196944
sage: a.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]

Python

>>> from sage.all import *
>>> a = sloane.A000153; a
a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(8))
82508
>>> a(Integer(20))
10315043624498196944
>>> a.list(Integer(8))
[0, 1, 2, 7, 32, 181, 1214, 9403]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000165

Bases: SloaneSequence

Double factorial numbers: \((2n)!! = 2^n*n!\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000165;a
Double factorial numbers: (2n)!! = 2^n*n!.
sage: a(0)
1
sage: a.offset
0
sage: a(8)
10321920
sage: a(20)
2551082656125828464640000
sage: a.list(9)
[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

Python

>>> from sage.all import *
>>> a = sloane.A000165;a
Double factorial numbers: (2n)!! = 2^n*n!.
>>> a(Integer(0))
1
>>> a.offset
0
>>> a(Integer(8))
10321920
>>> a(Integer(20))
2551082656125828464640000
>>> a.list(Integer(9))
[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

AUTHORS:

• Jaap Spies (2007-01-24)

class sage.combinat.sloane_functions.A000166

Bases: SloaneSequence

Subfactorial or rencontres numbers, or derangements: number of permutations of \(n\) elements with no fixed points.

With offset 1 also the permanent of a (0,1)-matrix of order \(n\) with \(n\) 0’s not on a line.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000166;a
Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.
sage: a(0)
1
sage: a(1)
0
sage: a(2)
1
sage: a.offset
0
sage: a(8)
14833
sage: a(20)
895014631192902121
sage: a.list(9)
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]

Python

>>> from sage.all import *
>>> a = sloane.A000166;a
Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.
>>> a(Integer(0))
1
>>> a(Integer(1))
0
>>> a(Integer(2))
1
>>> a.offset
0
>>> a(Integer(8))
14833
>>> a(Integer(20))
895014631192902121
>>> a.list(Integer(9))
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000169

Bases: SloaneSequence

Number of labeled rooted trees with \(n\) nodes: \(n^{(n-1)}\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000169;a
Number of labeled rooted trees with n nodes: n^(n-1).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(10)
1000000000
sage: a.list(11)
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]

Python

>>> from sage.all import *
>>> a = sloane.A000169;a
Number of labeled rooted trees with n nodes: n^(n-1).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(10))
1000000000
>>> a.list(Integer(11))
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000203

Bases: SloaneSequence

The sequence \(\sigma(n)\), where \(\sigma(n)\) is the sum of the divisors of \(n\). Also called \(\sigma_1(n)\).

The function sigma(n, k) implements \(\sigma_k(n)\) in Sage.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000203; a
sigma(n) = sum of divisors of n. Also called sigma_1(n).
sage: a(1)
1
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(256)
511
sage: a.list(12)
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a = sloane.A000203; a
sigma(n) = sum of divisors of n. Also called sigma_1(n).
>>> a(Integer(1))
1
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(256))
511
>>> a.list(Integer(12))
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000204

Bases: SloaneSequence

Lucas numbers (beginning with 1): \(L(n) = L(n-1) + L(n-2)\) with \(L(1) = 1\), \(L(2) = 3\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000204; a
Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A000204; a
Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
>>> a(Integer(1))
1
>>> a(Integer(8))
47
>>> a(Integer(200))
627376215338105766356982006981782561278127
>>> a(-Integer(4))
Traceback (most recent call last):
...
ValueError: input n (=-4) must be a positive integer
>>> a.list(Integer(12))
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-18)

class sage.combinat.sloane_functions.A000213

Bases: SloaneSequence

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, …

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000213;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
1
sage: a(11)
355
sage: a.list(12)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

Python

>>> from sage.all import *
>>> a = sloane.A000213;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(11))
355
>>> a.list(Integer(12))
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

AUTHORS:

• Jaap Spies (2007-01-19)

list(n)

EXAMPLES:

Sage

sage: sloane.A000213.list(10)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

Python

>>> from sage.all import *
>>> sloane.A000213.list(Integer(10))
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

class sage.combinat.sloane_functions.A000217

Bases: SloaneSequence

Triangular numbers: \(a(n) = \binom{n+1}{2} = n(n+1)/2\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000217;a
Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
sage: a(0)
0
sage: a(2)
3
sage: a(8)
36
sage: a(2000)
2001000
sage: a.list(9)
[0, 1, 3, 6, 10, 15, 21, 28, 36]

Python

>>> from sage.all import *
>>> a = sloane.A000217;a
Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
>>> a(Integer(0))
0
>>> a(Integer(2))
3
>>> a(Integer(8))
36
>>> a(Integer(2000))
2001000
>>> a.list(Integer(9))
[0, 1, 3, 6, 10, 15, 21, 28, 36]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000225

Bases: SloaneSequence

\(2^n - 1\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000225;a
2^n - 1.
sage: a(0)
0
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(12)
4095
sage: a.list(12)
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

Python

>>> from sage.all import *
>>> a = sloane.A000225;a
2^n - 1.
>>> a(Integer(0))
0
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(12))
4095
>>> a.list(Integer(12))
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000244

Bases: SloaneSequence

Powers of 3: \(a(n) = 3^n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000244;a
Powers of 3: a(n) = 3^n.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(3)
27
sage: a(11)
177147
sage: a.list(12)
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

Python

>>> from sage.all import *
>>> a = sloane.A000244;a
Powers of 3: a(n) = 3^n.
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
1
>>> a(Integer(3))
27
>>> a(Integer(11))
177147
>>> a.list(Integer(12))
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000255

Bases: ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-1)*a(n-2)\), with \(a(0) = 1\), \(a(1) = 1\).

With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=1\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000255;a
a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
sage: a(0)
1
sage: a(1)
1
sage: a.offset
0
sage: a(8)
148329
sage: a(22)
9923922230666898717143
sage: a.list(9)
[1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

Python

>>> from sage.all import *
>>> a = sloane.A000255;a
a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a.offset
0
>>> a(Integer(8))
148329
>>> a(Integer(22))
9923922230666898717143
>>> a.list(Integer(9))
[1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A000261

Bases: ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-3)*a(n-2)\), with \(a(1) = 1\), \(a(2) = 1\).

With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000261;a
a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a.offset
1
sage: a(8)
30637
sage: a(22)
1801366114380914335441
sage: a.list(9)
[0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

Python

>>> from sage.all import *
>>> a = sloane.A000261;a
a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
0
>>> a.offset
1
>>> a(Integer(8))
30637
>>> a(Integer(22))
1801366114380914335441
>>> a.list(Integer(9))
[0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A000272

Bases: SloaneSequence

Number of labeled rooted trees on \(n\) nodes: \(n^{(n-2)}\).

INPUT:

• n – integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000272;a
Number of labeled rooted trees with n nodes: n^(n-2).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
1
sage: a(10)
100000000
sage: a.list(12)
[1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

Python

>>> from sage.all import *
>>> a = sloane.A000272;a
Number of labeled rooted trees with n nodes: n^(n-2).
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(10))
100000000
>>> a.list(Integer(12))
[1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000290

Bases: SloaneSequence

The squares: \(a(n) = n^2\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000290;a
The squares: a(n) = n^2.
sage: a(0)
0
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(16)
256
sage: a.list(17)
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]

Python

>>> from sage.all import *
>>> a = sloane.A000290;a
The squares: a(n) = n^2.
>>> a(Integer(0))
0
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(16))
256
>>> a.list(Integer(17))
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000292

Bases: SloaneSequence

Tetrahedral (or pyramidal) numbers: \(\binom{n+2}{3} = n(n+1)(n+2)/6\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000292;a
Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
sage: a(0)
0
sage: a(2)
4
sage: a(11)
286
sage: a.list(12)
[0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

Python

>>> from sage.all import *
>>> a = sloane.A000292;a
Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
>>> a(Integer(0))
0
>>> a(Integer(2))
4
>>> a(Integer(11))
286
>>> a.list(Integer(12))
[0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000302

Bases: SloaneSequence

Powers of 4: \(a(n) = 4^n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000302;a
Powers of 4: a(n) = 4^n.
sage: a(0)
1
sage: a(1)
4
sage: a(2)
16
sage: a(10)
1048576
sage: a.list(12)
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

Python

>>> from sage.all import *
>>> a = sloane.A000302;a
Powers of 4: a(n) = 4^n.
>>> a(Integer(0))
1
>>> a(Integer(1))
4
>>> a(Integer(2))
16
>>> a(Integer(10))
1048576
>>> a.list(Integer(12))
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000312

Bases: SloaneSequence

Number of labeled mappings from \(n\) points to themselves (endofunctions): \(n^n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000312;a
Number of labeled mappings from n points to themselves (endofunctions): n^n.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(1)
1
sage: a(9)
387420489
sage: a.list(11)
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]

Python

>>> from sage.all import *
>>> a = sloane.A000312;a
Number of labeled mappings from n points to themselves (endofunctions): n^n.
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(9))
387420489
>>> a.list(Integer(11))
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000326

Bases: SloaneSequence

Pentagonal numbers: \(n(3n-1)/2\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000326;a
Pentagonal numbers: n(3n-1)/2.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
5
sage: a(10)
145
sage: a.list(12)
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a = sloane.A000326;a
Pentagonal numbers: n(3n-1)/2.
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
5
>>> a(Integer(10))
145
>>> a.list(Integer(12))
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000330

Bases: SloaneSequence

Square pyramidal numbers” \(0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000330;a
Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
14
sage: a(11)
506
sage: a.list(12)
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

Python

>>> from sage.all import *
>>> a = sloane.A000330;a
Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
0
>>> a(Integer(3))
14
>>> a(Integer(11))
506
>>> a.list(Integer(12))
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000396

Bases: SloaneSequence

Perfect numbers: equal to sum of proper divisors.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000396;a
Perfect numbers: equal to sum of proper divisors.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
28
sage: a(7)
137438691328
sage: a.list(7)
[6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

Python

>>> from sage.all import *
>>> a = sloane.A000396;a
Perfect numbers: equal to sum of proper divisors.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
6
>>> a(Integer(2))
28
>>> a(Integer(7))
137438691328
>>> a.list(Integer(7))
[6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000578

Bases: SloaneSequence

The cubes: \(a(n) = n^3\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000578;a
The cubes: n^3
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
27
sage: a(11)
1331
sage: a.list(12)
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

Python

>>> from sage.all import *
>>> a = sloane.A000578;a
The cubes: n^3
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
0
>>> a(Integer(3))
27
>>> a(Integer(11))
1331
>>> a.list(Integer(12))
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A000583

Bases: SloaneSequence

Fourth powers: \(a(n) = n^4\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000583;a
Fourth powers: n^4.
sage: a(0.0)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
sage: a(1)
1
sage: a(2)
16
sage: a(9)
6561
sage: a.list(10)
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

Python

>>> from sage.all import *
>>> a = sloane.A000583;a
Fourth powers: n^4.
>>> a(RealNumber('0.0'))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
>>> a(Integer(1))
1
>>> a(Integer(2))
16
>>> a(Integer(9))
6561
>>> a.list(Integer(10))
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

AUTHORS:

• Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A000587

Bases: ExponentialNumbers

The sequence of Uppuluri-Carpenter numbers.

The Uppuluri-Carpenter number \(C_n\) counts the imbalance in the number of ways to put \(n\) distinguishable things into an even number of indistinguishable boxes versus into an odd number of indistinguishable boxes, such that no box is empty.

Let \(S(n, k)\) denote the Stirling number of the second kind. Then

\[C_n = \sum{k=0}^{n} (-1)^k S(n, k) .\]

INPUT:

• n – non negative integer

OUTPUT:

• integer – \(C_n\)

EXAMPLES:

Sage

sage: a = sloane.A000587; a
Sequence of Uppuluri-Carpenter numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
sage: a.list(10)
[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

Python

>>> from sage.all import *
>>> a = sloane.A000587; a
Sequence of Uppuluri-Carpenter numbers
>>> a.offset
0
>>> a(Integer(0))
1
>>> a(Integer(100))
397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
>>> a.list(Integer(10))
[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

AUTHORS:

• Nick Alexander

class sage.combinat.sloane_functions.A000668

Bases: SloaneSequence

Mersenne primes (of form \(2^p - 1\) where \(p\) is a prime).

(See A000043 for the values of \(p\).)

Warning: a(39) has 4,053,946 digits!

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000668;a
Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
sage: a(1)
3
sage: a(2)
7
sage: a(12)
170141183460469231731687303715884105727

Python

>>> from sage.all import *
>>> a = sloane.A000668;a
Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
>>> a(Integer(1))
3
>>> a(Integer(2))
7
>>> a(Integer(12))
170141183460469231731687303715884105727

Warning: a(39) has 4,053,946 digits!

Sage

sage: a(40)
Traceback (most recent call last):
...
IndexError: list index out of range
sage: a.list(8)
[3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

Python

>>> from sage.all import *
>>> a(Integer(40))
Traceback (most recent call last):
...
IndexError: list index out of range
>>> a.list(Integer(8))
[3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000670

Bases: SloaneSequence

Number of preferential arrangements of \(n\) labeled elements; or number of weak orders on \(n\) labeled elements.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000670;a
Number of preferential arrangements of n labeled elements.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
3
sage: a(9)
7087261
sage: a.list(10)
[1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

Python

>>> from sage.all import *
>>> a = sloane.A000670;a
Number of preferential arrangements of n labeled elements.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(2))
3
>>> a(Integer(9))
7087261
>>> a.list(Integer(10))
[1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

AUTHORS:

• Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A000720

Bases: SloaneSequence

\(pi(n)\), the number of primes \(\le n\). Sometimes called \(PrimePi(n)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000720;a
pi(n), the number of primes <= n. Sometimes called PrimePi(n)
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a(8)
4
sage: a(1000)
168
sage: a.list(12)
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

Python

>>> from sage.all import *
>>> a = sloane.A000720;a
pi(n), the number of primes <= n. Sometimes called PrimePi(n)
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
1
>>> a(Integer(8))
4
>>> a(Integer(1000))
168
>>> a.list(Integer(12))
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A000796

Bases: SloaneSequence

Decimal expansion of \(\pi\).

INPUT:

• n – positive integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000796;a
Decimal expansion of Pi.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(13)
9
sage: a.list(14)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
sage: a(100)
7

Python

>>> from sage.all import *
>>> a = sloane.A000796;a
Decimal expansion of Pi.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
3
>>> a(Integer(13))
9
>>> a.list(Integer(14))
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
>>> a(Integer(100))
7

AUTHOR:

• Jaap Spies (2007-01-30)

list(n)

EXAMPLES:

Sage

sage: sloane.A000796.list(10)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

Python

>>> from sage.all import *
>>> sloane.A000796.list(Integer(10))
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

pi()

Based on an algorithm of Lambert Meertens The ABC-programming language!!!

EXAMPLES:

Sage

sage: it = sloane.A000796.pi()
sage: [next(it) for i in range(10)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

Python

>>> from sage.all import *
>>> it = sloane.A000796.pi()
>>> [next(it) for i in range(Integer(10))]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

class sage.combinat.sloane_functions.A000961

Bases: SloaneSequence

Prime powers

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000961;a
Prime powers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

Python

>>> from sage.all import *
>>> a = sloane.A000961;a
Prime powers.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
2
>>> a(Integer(12))
17
>>> a.list(Integer(12))
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

AUTHORS:

• Jaap Spies (2007-01-25)

list(n)

EXAMPLES:

Sage

sage: sloane.A000961.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

Python

>>> from sage.all import *
>>> sloane.A000961.list(Integer(10))
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

class sage.combinat.sloane_functions.A000984

Bases: SloaneSequence

Central binomial coefficients: \(\binom{2n}{n} = \frac {(2n)!} {(n!)^2}\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A000984;a
Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
sage: a(0)
1
sage: a(2)
6
sage: a(8)
12870
sage: a.list(9)
[1, 2, 6, 20, 70, 252, 924, 3432, 12870]

Python

>>> from sage.all import *
>>> a = sloane.A000984;a
Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
>>> a(Integer(0))
1
>>> a(Integer(2))
6
>>> a(Integer(8))
12870
>>> a.list(Integer(9))
[1, 2, 6, 20, 70, 252, 924, 3432, 12870]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001006

Bases: SloaneSequence

Motzkin numbers: number of ways of drawing any number of nonintersecting chords among \(n\) points on a circle.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001006;a
Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
2
sage: a(12)
15511
sage: a.list(13)
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]

Python

>>> from sage.all import *
>>> a = sloane.A001006;a
Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(12))
15511
>>> a.list(Integer(13))
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]

AUTHORS:

• Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A001045

Bases: RecurrenceSequence2

Jacobsthal sequence: \(a(n) = a(n-1) + 2a(n-2)\), \(a(0) = 0\) and \(a(1) = 1\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001045;a
Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
1
sage: a(11)
683
sage: a.list(12)
[0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

Python

>>> from sage.all import *
>>> a = sloane.A001045;a
Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(11))
683
>>> a.list(Integer(12))
[0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001055

Bases: SloaneSequence

Number of ways of factoring \(n\) with all factors 1.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001055;a
Number of ways of factoring n with all factors >1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(9)
2
sage: a.list(16)
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

Python

>>> from sage.all import *
>>> a = sloane.A001055;a
Number of ways of factoring n with all factors >1.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(9))
2
>>> a.list(Integer(16))
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

AUTHORS:

• Jaap Spies (2007-02-04)

nwf(n, m)

EXAMPLES:

Sage

sage: sloane.A001055.nwf(4,1)
0
sage: sloane.A001055.nwf(4,2)
1
sage: sloane.A001055.nwf(4,3)
1
sage: sloane.A001055.nwf(4,4)
2

Python

>>> from sage.all import *
>>> sloane.A001055.nwf(Integer(4),Integer(1))
0
>>> sloane.A001055.nwf(Integer(4),Integer(2))
1
>>> sloane.A001055.nwf(Integer(4),Integer(3))
1
>>> sloane.A001055.nwf(Integer(4),Integer(4))
2

class sage.combinat.sloane_functions.A001109

Bases: RecurrenceSequence2

\(a(n)^2\) is a triangular number: \(a(n) = 6*a(n-1) - a(n-2)\) with \(a(0)=0\), \(a(1)=1\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001109;a
a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
sage: a(0)
0
sage: a(1)
1
sage: a(2)
6
sage: a.offset
0
sage: a(8)
235416
sage: a(60)
1515330104844857898115857393785728383101709300
sage: a.list(9)
[0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

Python

>>> from sage.all import *
>>> a = sloane.A001109;a
a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
6
>>> a.offset
0
>>> a(Integer(8))
235416
>>> a(Integer(60))
1515330104844857898115857393785728383101709300
>>> a.list(Integer(9))
[0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

AUTHORS:

• Jaap Spies (2007-01-24)

class sage.combinat.sloane_functions.A001110

Bases: RecurrenceSequence

Numbers that are both triangular and square: \(a(n) = 34a(n-1) - a(n-2) + 2\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001110; a
Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
sage: a(0)
0
sage: a(1)
1
sage: a(8)
55420693056
sage: a(21)
4446390382511295358038307980025
sage: a.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

Python

>>> from sage.all import *
>>> a = sloane.A001110; a
Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(8))
55420693056
>>> a(Integer(21))
4446390382511295358038307980025
>>> a.list(Integer(8))
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

AUTHORS:

• Jaap Spies (2007-01-19)

g(k)

EXAMPLES:

Sage

sage: sloane.A001110.g(2)
2
sage: sloane.A001110.g(1)
0

Python

>>> from sage.all import *
>>> sloane.A001110.g(Integer(2))
2
>>> sloane.A001110.g(Integer(1))
0

link = 'http://oeis.org/classic/A001110'

class sage.combinat.sloane_functions.A001147

Bases: SloaneSequence

Double factorial numbers: \((2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001147;a
Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
sage: a(0)
1
sage: a.offset
0
sage: a(8)
2027025
sage: a(20)
319830986772877770815625
sage: a.list(9)
[1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

Python

>>> from sage.all import *
>>> a = sloane.A001147;a
Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
>>> a(Integer(0))
1
>>> a.offset
0
>>> a(Integer(8))
2027025
>>> a(Integer(20))
319830986772877770815625
>>> a.list(Integer(9))
[1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

AUTHORS:

• Jaap Spies (2007-01-24)

class sage.combinat.sloane_functions.A001157

Bases: SloaneSequence

The sequence \(\sigma_2(n)\), sum of squares of divisors of \(n\).

The function sigma(n, k) implements \(\sigma_k*\) in Sage.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001157;a
sigma_2(n): sum of squares of divisors of n
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
5
sage: a(8)
85
sage: a.list(9)
[1, 5, 10, 21, 26, 50, 50, 85, 91]

Python

>>> from sage.all import *
>>> a = sloane.A001157;a
sigma_2(n): sum of squares of divisors of n
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
5
>>> a(Integer(8))
85
>>> a.list(Integer(9))
[1, 5, 10, 21, 26, 50, 50, 85, 91]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A001189

Bases: SloaneSequence

Number of degree-n permutations of order exactly 2.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001189;a
Number of degree-n permutations of order exactly 2.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(2)
1
sage: a(12)
140151
sage: a.list(13)
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

Python

>>> from sage.all import *
>>> a = sloane.A001189;a
Number of degree-n permutations of order exactly 2.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
0
>>> a(Integer(2))
1
>>> a(Integer(12))
140151
>>> a.list(Integer(13))
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

AUTHORS:

• Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A001221

Bases: SloaneSequence

Number of different prime divisors of \(n\)

Also called omega(n) or \(\omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001221; a
Number of distinct primes dividing n (also called omega(n)).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
1
sage: a(41)
1
sage: a(84792)
3
sage: a.list(12)
[0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

Python

>>> from sage.all import *
>>> a = sloane.A001221; a
Number of distinct primes dividing n (also called omega(n)).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
0
>>> a(Integer(8))
1
>>> a(Integer(41))
1
>>> a(Integer(84792))
3
>>> a.list(Integer(12))
[0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A001222

Bases: SloaneSequence

Number of prime divisors of \(n\) (counted with multiplicity).

Also called bigomega(n) or \(\Omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001222; a
Number of prime divisors of n (counted with multiplicity).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
3
sage: a(41)
1
sage: a(84792)
5
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

Python

>>> from sage.all import *
>>> a = sloane.A001222; a
Number of prime divisors of n (counted with multiplicity).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
0
>>> a(Integer(8))
3
>>> a(Integer(41))
1
>>> a(Integer(84792))
5
>>> a.list(Integer(12))
[0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A001227

Bases: SloaneSequence

Number of odd divisors of \(n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001227; a
Number of odd divisors of n
sage: a.offset
1
sage: a(1)
1
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
3
sage: a(256)
1
sage: a(29)
2
sage: a.list(20)
[1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A001227; a
Number of odd divisors of n
>>> a.offset
1
>>> a(Integer(1))
1
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(100))
3
>>> a(Integer(256))
1
>>> a(Integer(29))
2
>>> a.list(Integer(20))
[1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-14)

class sage.combinat.sloane_functions.A001333

Bases: RecurrenceSequence2

Numerators of continued fraction convergents to \(\sqrt 2\).

See also A000129

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001333;a
Numerators of continued fraction convergents to sqrt(2).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
3
sage: a(3)
7
sage: a(11)
8119
sage: a.list(12)
[1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

Python

>>> from sage.all import *
>>> a = sloane.A001333;a
Numerators of continued fraction convergents to sqrt(2).
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(2))
3
>>> a(Integer(3))
7
>>> a(Integer(11))
8119
>>> a.list(Integer(12))
[1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

AUTHORS:

• Jaap Spies (2007-02-01)

class sage.combinat.sloane_functions.A001358

Bases: SloaneSequence

Products of two primes.

These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001358;a
Products of two primes.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(8)
22
sage: a(200)
669
sage: a.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]

Python

>>> from sage.all import *
>>> a = sloane.A001358;a
Products of two primes.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
6
>>> a(Integer(8))
22
>>> a(Integer(200))
669
>>> a.list(Integer(9))
[4, 6, 9, 10, 14, 15, 21, 22, 25]

AUTHORS:

• Jaap Spies (2007-01-25)

list(n)

EXAMPLES:

Sage

sage: sloane.A001358.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]

Python

>>> from sage.all import *
>>> sloane.A001358.list(Integer(9))
[4, 6, 9, 10, 14, 15, 21, 22, 25]

class sage.combinat.sloane_functions.A001405

Bases: SloaneSequence

Central binomial coefficients: \(\binom{n}{\lfloor \frac {n}{ 2} \rfloor}\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001405;a
Central binomial coefficients: C(n,floor(n/2)).
sage: a(0)
1
sage: a(2)
2
sage: a(12)
924
sage: a.list(12)
[1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

Python

>>> from sage.all import *
>>> a = sloane.A001405;a
Central binomial coefficients: C(n,floor(n/2)).
>>> a(Integer(0))
1
>>> a(Integer(2))
2
>>> a(Integer(12))
924
>>> a.list(Integer(12))
[1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001477

Bases: SloaneSequence

The nonnegative integers.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001477;a
The nonnegative integers.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3382789)
3382789
sage: a(11)
11
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Python

>>> from sage.all import *
>>> a = sloane.A001477;a
The nonnegative integers.
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
0
>>> a(Integer(3382789))
3382789
>>> a(Integer(11))
11
>>> a.list(Integer(12))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A001694

Bases: SloaneSequence

This function returns the \(n\)-th Powerful Number:

A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001694; a
Powerful Numbers (also called squarefull, square-full or 2-full numbers).
sage: a.offset
1
sage: a(1)
1
sage: a(4)
9
sage: a(100)
3136
sage: a(156)
7225
sage: a.list(19)
[1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A001694; a
Powerful Numbers (also called squarefull, square-full or 2-full numbers).
>>> a.offset
1
>>> a(Integer(1))
1
>>> a(Integer(4))
9
>>> a(Integer(100))
3136
>>> a(Integer(156))
7225
>>> a.list(Integer(19))
[1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-14)

is_powerful(n)

Return True if and only if \(n\) is a powerful number.

A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\).

See OEIS sequence A001694.

INPUT:

• \(n\) – integer

OUTPUT:

True if \(n\) is a powerful number, else False

EXAMPLES:

Sage

sage: a = sloane.A001694
sage: a.is_powerful(2500)
True
sage: a.is_powerful(20)
False

Python

>>> from sage.all import *
>>> a = sloane.A001694
>>> a.is_powerful(Integer(2500))
True
>>> a.is_powerful(Integer(20))
False

AUTHORS:

• Jaap Spies (2006-12-07)

list(n)

EXAMPLES:

Sage

sage: sloane.A001694.list(9)
[1, 4, 8, 9, 16, 25, 27, 32, 36]

Python

>>> from sage.all import *
>>> sloane.A001694.list(Integer(9))
[1, 4, 8, 9, 16, 25, 27, 32, 36]

class sage.combinat.sloane_functions.A001836

Bases: SloaneSequence

Numbers \(n\) such that \(\phi(2n-1) < \phi(2n)\), where \(\phi\) is Euler’s totient function.

Euler’s totient function is also known as euler_phi, euler_phi is a standard Sage function.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001836; a
Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
sage: a.offset
1
sage: a(1)
53
sage: a(8)
683
sage: a(300)
17798
sage: a.list(12)
[53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A001836; a
Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
>>> a.offset
1
>>> a(Integer(1))
53
>>> a(Integer(8))
683
>>> a(Integer(300))
17798
>>> a.list(Integer(12))
[53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Compare: Searching Sloane’s online database… Numbers n such that phi(2n-1) phi(2n), where phi is Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]

AUTHORS:

• Jaap Spies (2007-01-17)

list(n)

EXAMPLES:

Sage

sage: sloane.A001836.list(9)
[53, 83, 158, 263, 293, 368, 578, 683, 743]

Python

>>> from sage.all import *
>>> sloane.A001836.list(Integer(9))
[53, 83, 158, 263, 293, 368, 578, 683, 743]

class sage.combinat.sloane_functions.A001906

Bases: RecurrenceSequence2

\(F(2n) =\) bisection of Fibonacci sequence: \(a(n)=3a(n-1)-a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001906; a
F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
987
sage: a(22)
701408733
sage: a.list(12)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

Python

>>> from sage.all import *
>>> a = sloane.A001906; a
F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(8))
987
>>> a(Integer(22))
701408733
>>> a.list(Integer(12))
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A001909

Bases: ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-4)*a(n-2)\), with \(a(2) = 0\), \(a(3) = 1\).

With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – positive integer >= 2

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001909;a
a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
sage: a(1)
Traceback (most recent call last):
...
ValueError: input n (=1) must be an integer >= 2
sage: a.offset
2
sage: a(2)
0
sage: a(8)
8544
sage: a(22)
470033715095287415734
sage: a.list(9)
[0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

Python

>>> from sage.all import *
>>> a = sloane.A001909;a
a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
>>> a(Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=1) must be an integer >= 2
>>> a.offset
2
>>> a(Integer(2))
0
>>> a(Integer(8))
8544
>>> a(Integer(22))
470033715095287415734
>>> a.list(Integer(9))
[0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A001910

Bases: ExtremesOfPermanentsSequence

\(a(n) = n*a(n-1) + (n-5)*a(n-2)\), with \(a(3) = 0\), \(a(4) = 1\).

With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – positive integer >= 3

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001910;a
a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be an integer >= 3
sage: a(3)
0
sage: a.offset
3
sage: a(8)
1909
sage: a(22)
98125321641110663023
sage: a.list(9)
[0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]

Python

>>> from sage.all import *
>>> a = sloane.A001910;a
a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be an integer >= 3
>>> a(Integer(3))
0
>>> a.offset
3
>>> a(Integer(8))
1909
>>> a(Integer(22))
98125321641110663023
>>> a.list(Integer(9))
[0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A001969

Bases: SloaneSequence

Evil numbers: even number of 1’s in binary expansion.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A001969;a
Evil numbers: even number of 1's in binary expansion.
sage: a(0)
0
sage: a(1)
3
sage: a(2)
5
sage: a(12)
24
sage: a.list(13)
[0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

Python

>>> from sage.all import *
>>> a = sloane.A001969;a
Evil numbers: even number of 1's in binary expansion.
>>> a(Integer(0))
0
>>> a(Integer(1))
3
>>> a(Integer(2))
5
>>> a(Integer(12))
24
>>> a.list(Integer(13))
[0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

AUTHORS:

• Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A002110

Bases: SloaneSequence

Primorial numbers (first definition): product of first \(n\) primes. Sometimes written \(p\#\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A002110;a
Primorial numbers (first definition): product of first n primes. Sometimes written p#.
sage: a(0)
1
sage: a(2)
6
sage: a(8)
9699690
sage: a(17)
1922760350154212639070
sage: a.list(9)
[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

Python

>>> from sage.all import *
>>> a = sloane.A002110;a
Primorial numbers (first definition): product of first n primes. Sometimes written p#.
>>> a(Integer(0))
1
>>> a(Integer(2))
6
>>> a(Integer(8))
9699690
>>> a(Integer(17))
1922760350154212639070
>>> a.list(Integer(9))
[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A002113

Bases: SloaneSequence

Palindromes in base 10.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A002113;a
Palindromes in base 10.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
2
sage: a(12)
33
sage: a.list(13)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

Python

>>> from sage.all import *
>>> a = sloane.A002113;a
Palindromes in base 10.
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(12))
33
>>> a.list(Integer(13))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

AUTHORS:

• Jaap Spies (2007-02-02)

list(n)

EXAMPLES:

Sage

sage: sloane.A002113.list(15)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]

Python

>>> from sage.all import *
>>> sloane.A002113.list(Integer(15))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]

class sage.combinat.sloane_functions.A002275

Bases: SloaneSequence

Repunits: \(\frac {(10^n - 1)}{9}\). Often denoted by \(R_n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A002275;a
Repunits: (10^n - 1)/9. Often denoted by R_n.
sage: a(0)
0
sage: a(2)
11
sage: a(8)
11111111
sage: a(20)
11111111111111111111
sage: a.list(9)
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

Python

>>> from sage.all import *
>>> a = sloane.A002275;a
Repunits: (10^n - 1)/9. Often denoted by R_n.
>>> a(Integer(0))
0
>>> a(Integer(2))
11
>>> a(Integer(8))
11111111
>>> a(Integer(20))
11111111111111111111
>>> a.list(Integer(9))
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A002378

Bases: SloaneSequence

Oblong (or pronic, or heteromecic) numbers: \(n(n+1)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A002378;a
Oblong (or pronic, or heteromecic) numbers: n(n+1).
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(1)
2
sage: a(11)
132
sage: a.list(12)
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

Python

>>> from sage.all import *
>>> a = sloane.A002378;a
Oblong (or pronic, or heteromecic) numbers: n(n+1).
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
0
>>> a(Integer(1))
2
>>> a(Integer(11))
132
>>> a.list(Integer(12))
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A002620

Bases: SloaneSequence

Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, \(\lfloor n^2/4 \rfloor\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A002620;a
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(10)
25
sage: a.list(12)
[0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

Python

>>> from sage.all import *
>>> a = sloane.A002620;a
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
>>> a(Integer(0))
0
>>> a(Integer(1))
0
>>> a(Integer(2))
1
>>> a(Integer(10))
25
>>> a.list(Integer(12))
[0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A002808

Bases: SloaneSequence

The composite numbers: numbers \(n\) of the form \(xy\) for \(x > 1\) and \(y > 1\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A002808;a
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(11)
20
sage: a.list(12)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

Python

>>> from sage.all import *
>>> a = sloane.A002808;a
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
6
>>> a(Integer(11))
20
>>> a.list(Integer(12))
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

AUTHORS:

• Jaap Spies (2007-01-26)

list(n)

EXAMPLES:

Sage

sage: sloane.A002808.list(10)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

Python

>>> from sage.all import *
>>> sloane.A002808.list(Integer(10))
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

class sage.combinat.sloane_functions.A003418

Bases: SloaneSequence

Least common multiple (or lcm) of \(\{1, 2, \ldots, n\}\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A003418;a
Least common multiple (or lcm) of {1, 2, ..., n}.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
360360
sage: a.list(14)
[1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
sage: a(20.0)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a = sloane.A003418;a
Least common multiple (or lcm) of {1, 2, ..., n}.
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(13))
360360
>>> a.list(Integer(14))
[1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
>>> a(RealNumber('20.0'))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHOR:

• Jaap Spies (2007-01-31)

class sage.combinat.sloane_functions.A004086

Bases: SloaneSequence

Read n backwards (referred to as \(R(n)\) in many sequences).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A004086;a
Read n backwards (referred to as R(n) in many sequences).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
2
sage: a(3333)
3333
sage: a(12345)
54321
sage: a.list(13)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

Python

>>> from sage.all import *
>>> a = sloane.A004086;a
Read n backwards (referred to as R(n) in many sequences).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(3333))
3333
>>> a(Integer(12345))
54321
>>> a.list(Integer(13))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

AUTHORS:

• Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A004526

Bases: SloaneSequence

The nonnegative integers repeated.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A004526;a
The nonnegative integers repeated.
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(10)
5
sage: a.list(12)
[0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

Python

>>> from sage.all import *
>>> a = sloane.A004526;a
The nonnegative integers repeated.
>>> a(Integer(0))
0
>>> a(Integer(1))
0
>>> a(Integer(2))
1
>>> a(Integer(10))
5
>>> a.list(Integer(12))
[0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A005100

Bases: SloaneSequence

Deficient numbers: \(\sigma(n) < 2n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A005100;a
Deficient numbers: sigma(n) < 2n
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(12)
14
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

Python

>>> from sage.all import *
>>> a = sloane.A005100;a
Deficient numbers: sigma(n) < 2n
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(12))
14
>>> a.list(Integer(12))
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

AUTHORS:

• Jaap Spies (2007-01-26)

list(n)

EXAMPLES:

Sage

sage: sloane.A005100.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]

Python

>>> from sage.all import *
>>> sloane.A005100.list(Integer(10))
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]

class sage.combinat.sloane_functions.A005101

Bases: SloaneSequence

Abundant numbers (sum of divisors of \(n\) exceeds \(2n\)).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A005101;a
Abundant numbers (sum of divisors of n exceeds 2n).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
12
sage: a(2)
18
sage: a(12)
60
sage: a.list(12)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

Python

>>> from sage.all import *
>>> a = sloane.A005101;a
Abundant numbers (sum of divisors of n exceeds 2n).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
12
>>> a(Integer(2))
18
>>> a(Integer(12))
60
>>> a.list(Integer(12))
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

AUTHORS:

• Jaap Spies (2007-01-26)

list(n)

EXAMPLES:

Sage

sage: sloane.A005101.list(10)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

Python

>>> from sage.all import *
>>> sloane.A005101.list(Integer(10))
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

class sage.combinat.sloane_functions.A005117

Bases: SloaneSequence

Square-free numbers

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A005117;a
Square-free numbers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

Python

>>> from sage.all import *
>>> a = sloane.A005117;a
Square-free numbers.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
2
>>> a(Integer(12))
17
>>> a.list(Integer(12))
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

AUTHORS:

• Jaap Spies (2007-01-25)

list(n)

EXAMPLES:

Sage

sage: sloane.A005117.list(10)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

Python

>>> from sage.all import *
>>> sloane.A005117.list(Integer(10))
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

class sage.combinat.sloane_functions.A005408

Bases: SloaneSequence

The odd numbers a(n) = 2n + 1.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A005408;a
The odd numbers a(n) = 2n + 1.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(4)
9
sage: a(11)
23
sage: a.list(12)
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

Python

>>> from sage.all import *
>>> a = sloane.A005408;a
The odd numbers a(n) = 2n + 1.
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
>>> a(Integer(0))
1
>>> a(Integer(4))
9
>>> a(Integer(11))
23
>>> a.list(Integer(12))
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

AUTHORS:

• Jaap Spies (2007-01-26)

class sage.combinat.sloane_functions.A005843

Bases: SloaneSequence

The even numbers: \(a(n) = 2n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A005843;a
The even numbers: a(n) = 2n.
sage: a(0.0)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
sage: a(1)
2
sage: a(2)
4
sage: a(9)
18
sage: a.list(10)
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

Python

>>> from sage.all import *
>>> a = sloane.A005843;a
The even numbers: a(n) = 2n.
>>> a(RealNumber('0.0'))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
>>> a(Integer(1))
2
>>> a(Integer(2))
4
>>> a(Integer(9))
18
>>> a.list(Integer(10))
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

AUTHORS:

• Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A006318

Bases: SloaneSequence

Large Schroeder numbers.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A006318;a
Large Schroeder numbers.
sage: a(0)
1
sage: a(1)
2
sage: a(2)
6
sage: a(9)
206098
sage: a.list(10)
[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

Python

>>> from sage.all import *
>>> a = sloane.A006318;a
Large Schroeder numbers.
>>> a(Integer(0))
1
>>> a(Integer(1))
2
>>> a(Integer(2))
6
>>> a(Integer(9))
206098
>>> a.list(Integer(10))
[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

AUTHORS:

• Jaap Spies (2007-02-03)

class sage.combinat.sloane_functions.A006530

Bases: SloaneSequence

Largest prime dividing \(n\) (with \(a(1)=1\)).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A006530;a
Largest prime dividing n (with a(1)=1).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(8)
2
sage: a(11)
11
sage: a.list(15)
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

Python

>>> from sage.all import *
>>> a = sloane.A006530;a
Largest prime dividing n (with a(1)=1).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(8))
2
>>> a(Integer(11))
11
>>> a.list(Integer(15))
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

AUTHORS:

• Jaap Spies (2007-01-25)

class sage.combinat.sloane_functions.A006882

Bases: SloaneSequence

Double factorials \(n!!\): \(a(n)=n \cdot a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A006882;a
Double factorials n!!: a(n)=n*a(n-2).
sage: a(0)
1
sage: a(2)
2
sage: a(8)
384
sage: a(20)
3715891200
sage: a.list(9)
[1, 1, 2, 3, 8, 15, 48, 105, 384]

Python

>>> from sage.all import *
>>> a = sloane.A006882;a
Double factorials n!!: a(n)=n*a(n-2).
>>> a(Integer(0))
1
>>> a(Integer(2))
2
>>> a(Integer(8))
384
>>> a(Integer(20))
3715891200
>>> a.list(Integer(9))
[1, 1, 2, 3, 8, 15, 48, 105, 384]

AUTHORS:

• Jaap Spies (2007-01-24)

df()

Double factorials n!!: a(n)=n*a(n-2).

EXAMPLES:

Sage

sage: it = sloane.A006882.df()
sage: [next(it) for i in range(10)]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

Python

>>> from sage.all import *
>>> it = sloane.A006882.df()
>>> [next(it) for i in range(Integer(10))]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

list(n)

EXAMPLES:

Sage

sage: sloane.A006882.list(10)
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

Python

>>> from sage.all import *
>>> sloane.A006882.list(Integer(10))
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

class sage.combinat.sloane_functions.A007318

Bases: SloaneSequence

Pascal’s triangle read by rows: \(C(n,k) = \binom{n}{k} = \frac {n!} {(k!(n-k)!)}\), \(0 \le k \le n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A007318
sage: a(0)
1
sage: a(1)
1
sage: a(13)
4
sage: a.list(15)
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
sage: a(100)
715

Python

>>> from sage.all import *
>>> a = sloane.A007318
>>> a(Integer(0))
1
>>> a(Integer(1))
1
>>> a(Integer(13))
4
>>> a.list(Integer(15))
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
>>> a(Integer(100))
715

AUTHORS:

• Jaap Spies (2007-01-31)

keyword = ['nonn', 'tabl', 'nice', 'easy', 'core', 'triangle']

class sage.combinat.sloane_functions.A008275

Bases: SloaneSequence

Triangle of Stirling numbers of first kind, \(s(n,k)\), \(n \ge 1\), \(1 \le k \le n\).

The unsigned numbers are also called Stirling cycle numbers:

\(|s(n,k)|\) = number of permutations of \(n\) objects with exactly \(k\) cycles.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A008275;a
Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
-1
sage: a(3)
1
sage: a(11)
24
sage: a.list(12)
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

Python

>>> from sage.all import *
>>> a = sloane.A008275;a
Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
-1
>>> a(Integer(3))
1
>>> a(Integer(11))
24
>>> a.list(Integer(12))
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

AUTHORS:

• Jaap Spies (2007-02-02)

keyword = ['sign', 'tabl', 'nice', 'core', 'triangle']

s(n, k)

EXAMPLES:

Sage

sage: sloane.A008275.s(4,2)
11
sage: sloane.A008275.s(5,2)
-50
sage: sloane.A008275.s(5,3)
35

Python

>>> from sage.all import *
>>> sloane.A008275.s(Integer(4),Integer(2))
11
>>> sloane.A008275.s(Integer(5),Integer(2))
-50
>>> sloane.A008275.s(Integer(5),Integer(3))
35

class sage.combinat.sloane_functions.A008277

Bases: SloaneSequence

Triangle of Stirling numbers of 2nd kind, \(S2(n,k)\), \(n \ge 1\), \(1 \le k \le n\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A008277;a
Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(3)
1
sage: a(4.0)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
sage: a.list(15)
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

Python

>>> from sage.all import *
>>> a = sloane.A008277;a
Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(3))
1
>>> a(RealNumber('4.0'))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer
>>> a.list(Integer(15))
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

AUTHORS:

• Jaap Spies (2007-01-31)

keyword = ['nonn', 'tabl', 'nice', 'core', 'triangle']

s2(n, k)

Returns the Stirling number S2(n,k) of the 2nd kind.

EXAMPLES:

Sage

sage: sloane.A008277.s2(4,2)
7

Python

>>> from sage.all import *
>>> sloane.A008277.s2(Integer(4),Integer(2))
7

class sage.combinat.sloane_functions.A008683

Bases: SloaneSequence

Möbius function \(\mu(n)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A008683;a
Moebius function mu(n).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
-1
sage: a(12)
0
sage: a.list(12)
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

Python

>>> from sage.all import *
>>> a = sloane.A008683;a
Moebius function mu(n).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
-1
>>> a(Integer(12))
0
>>> a.list(Integer(12))
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A010060

Bases: SloaneSequence

Thue-Morse sequence.

Let \(A_k\) denote the first \(2^k\) terms; then \(A_0 = 0\), and for \(k \ge 0\), \(A_{k+1} = A_k B_k\), where \(B_k\) is obtained from \(A_k\) by interchanging 0’s and 1’s.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A010060;a
Thue-Morse sequence.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
1
sage: a(12)
0
sage: a.list(13)
[0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]

Python

>>> from sage.all import *
>>> a = sloane.A010060;a
Thue-Morse sequence.
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
1
>>> a(Integer(12))
0
>>> a.list(Integer(13))
[0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]

AUTHORS:

• Jaap Spies (2007-02-02)

class sage.combinat.sloane_functions.A015521

Bases: RecurrenceSequence2

Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 4 a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A015521; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
13107
sage: a(41)
967140655691703339764941
sage: a.list(12)
[0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

Python

>>> from sage.all import *
>>> a = sloane.A015521; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(8))
13107
>>> a(Integer(41))
967140655691703339764941
>>> a.list(Integer(12))
[0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015523

Bases: RecurrenceSequence2

Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 5 a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A015523; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
17727
sage: a(41)
6173719566474529739091481
sage: a.list(12)
[0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

Python

>>> from sage.all import *
>>> a = sloane.A015523; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(8))
17727
>>> a(Integer(41))
6173719566474529739091481
>>> a.list(Integer(12))
[0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015530

Bases: RecurrenceSequence2

Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 3 a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A015530;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
4
sage: a.offset
0
sage: a(8)
41008
sage: a.list(9)
[0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

Python

>>> from sage.all import *
>>> a = sloane.A015530;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
4
>>> a.offset
0
>>> a(Integer(8))
41008
>>> a.list(Integer(9))
[0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015531

Bases: RecurrenceSequence2

Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 5 a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A015531;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
4
sage: a.offset
0
sage: a(8)
65104
sage: a(60)
144560289664733924534327040115992228190104
sage: a.list(9)
[0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

Python

>>> from sage.all import *
>>> a = sloane.A015531;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
4
>>> a.offset
0
>>> a(Integer(8))
65104
>>> a(Integer(60))
144560289664733924534327040115992228190104
>>> a.list(Integer(9))
[0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A015551

Bases: RecurrenceSequence2

Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 6 a(n-1) + 5 a(n-2)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A015551;a
Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
6
sage: a.offset
0
sage: a(8)
570216
sage: a(60)
7110606606530059736761484557155863822531970573036
sage: a.list(9)
[0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

Python

>>> from sage.all import *
>>> a = sloane.A015551;a
Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
>>> a(Integer(0))
0
>>> a(Integer(1))
1
>>> a(Integer(2))
6
>>> a.offset
0
>>> a(Integer(8))
570216
>>> a(Integer(60))
7110606606530059736761484557155863822531970573036
>>> a.list(Integer(9))
[0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A018252

Bases: SloaneSequence

The nonprime numbers, starting with 1.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A018252;a
The nonprime numbers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
4
sage: a(9)
15
sage: a.list(10)
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

Python

>>> from sage.all import *
>>> a = sloane.A018252;a
The nonprime numbers.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
4
>>> a(Integer(9))
15
>>> a.list(Integer(10))
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

AUTHORS:

• Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A020639

Bases: SloaneSequence

Least prime dividing \(n\) with \(a(1)=1\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A020639;a
Least prime dividing n (a(1)=1).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(13)
13
sage: a.list(14)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

Python

>>> from sage.all import *
>>> a = sloane.A020639;a
Least prime dividing n (a(1)=1).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(13))
13
>>> a.list(Integer(14))
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

AUTHORS:

• Jaap Spies (2007-01-25)

list(n)

EXAMPLES:

Sage

sage: sloane.A020639.list(10)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

Python

>>> from sage.all import *
>>> sloane.A020639.list(Integer(10))
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

class sage.combinat.sloane_functions.A046660(offset=1)

Bases: SloaneSequence

Excess of \(n\) = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

\(\Omega(n) - \omega(n)\).

INPUT:

• n – positive integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A046660; a
Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
2
sage: a(41)
0
sage: a(84792)
2
sage: a.list(12)
[0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

Python

>>> from sage.all import *
>>> a = sloane.A046660; a
Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
0
>>> a(Integer(8))
2
>>> a(Integer(41))
0
>>> a(Integer(84792))
2
>>> a.list(Integer(12))
[0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A049310

Bases: SloaneSequence

Triangle of coefficients of Chebyshev’s \(S(n,x)\): \(U(n, \frac x 2)\) polynomials (exponents in increasing order).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A049310;a
Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
sage: a(0)
1
sage: a(1)
0
sage: a(13)
0
sage: a.list(15)
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
sage: a(200)
0
sage: a.keyword
['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

Python

>>> from sage.all import *
>>> a = sloane.A049310;a
Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
>>> a(Integer(0))
1
>>> a(Integer(1))
0
>>> a(Integer(13))
0
>>> a.list(Integer(15))
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
>>> a(Integer(200))
0
>>> a.keyword
['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

AUTHORS:

• Jaap Spies (2007-01-31)

keyword = ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

class sage.combinat.sloane_functions.A051959

Bases: RecurrenceSequence

Linear second order recurrence. A051959.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A051959; a
Linear second order recurrence. A051959.
sage: a(0)
1
sage: a(1)
10
sage: a(8)
9969
sage: a(41)
42834431872413650
sage: a.list(12)
[1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

Python

>>> from sage.all import *
>>> a = sloane.A051959; a
Linear second order recurrence. A051959.
>>> a(Integer(0))
1
>>> a(Integer(1))
10
>>> a(Integer(8))
9969
>>> a(Integer(41))
42834431872413650
>>> a.list(Integer(12))
[1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

AUTHORS:

• Jaap Spies (2007-01-19)

g(k)

EXAMPLES:

Sage

sage: sloane.A051959.g(2)
15
sage: sloane.A051959.g(1)
0

Python

>>> from sage.all import *
>>> sloane.A051959.g(Integer(2))
15
>>> sloane.A051959.g(Integer(1))
0

class sage.combinat.sloane_functions.A055790

Bases: ExtremesOfPermanentsSequence2

\(a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]\).

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A055790;a
a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
sage: a(0)
0
sage: a(1)
2
sage: a(2)
4
sage: a.offset
0
sage: a(8)
165016
sage: a(22)
10356214297533070441564
sage: a.list(9)
[0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

Python

>>> from sage.all import *
>>> a = sloane.A055790;a
a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
>>> a(Integer(0))
0
>>> a(Integer(1))
2
>>> a(Integer(2))
4
>>> a.offset
0
>>> a(Integer(8))
165016
>>> a(Integer(22))
10356214297533070441564
>>> a.list(Integer(9))
[0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A061084

Bases: SloaneSequence

Fibonacci-type sequence based on subtraction: \(a(0) = 1\), \(a(1) = 2\) and \(a(n) = a(n-2)-a(n-1)\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A061084; a
Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
sage: a(0)
1
sage: a(1)
2
sage: a(8)
-29
sage: a(22)
-24476
sage: a.list(12)
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
sage: a.keyword
['sign', 'easy', 'nice']

Python

>>> from sage.all import *
>>> a = sloane.A061084; a
Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
>>> a(Integer(0))
1
>>> a(Integer(1))
2
>>> a(Integer(8))
-29
>>> a(Integer(22))
-24476
>>> a.list(Integer(12))
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
>>> a.keyword
['sign', 'easy', 'nice']

AUTHORS:

• Jaap Spies (2007-01-18)

keyword = ['sign', 'easy', 'nice']

class sage.combinat.sloane_functions.A064553

Bases: SloaneSequence

\(a(1) = 1\), \(a(prime(i)) = i + 1\) for \(i > 0\) and \(a(u \cdot v) = a(u) \cdot a(v)\) for \(u, v > 0\).

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A064553;a
a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(9)
9
sage: a.list(16)
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

Python

>>> from sage.all import *
>>> a = sloane.A064553;a
a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
1
>>> a(Integer(2))
2
>>> a(Integer(9))
9
>>> a.list(Integer(16))
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

AUTHORS:

• Jaap Spies (2007-02-04)

class sage.combinat.sloane_functions.A079922(offset=1)

Bases: SloaneSequence

function returns solutions to the Dancing School problem with \(n\) girls and \(n+3\) boys.

The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\).

REFERENCES:

• Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

INPUT:

• n – positive integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A079922; a
Solutions to the Dancing School problem with n girls and n+3 boys
sage: a.offset
1
sage: a(1)
4
sage: a(8)
2227
sage: a.list(8)
[4, 13, 36, 90, 212, 478, 1044, 2227]

Python

>>> from sage.all import *
>>> a = sloane.A079922; a
Solutions to the Dancing School problem with n girls and n+3 boys
>>> a.offset
1
>>> a(Integer(1))
4
>>> a(Integer(8))
2227
>>> a.list(Integer(8))
[4, 13, 36, 90, 212, 478, 1044, 2227]

Compare: Searching Sloane’s online database… Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227]

Sage

sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Python

>>> from sage.all import *
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-14)

class sage.combinat.sloane_functions.A079923(offset=1)

Bases: SloaneSequence

function returns solutions to the Dancing School problem with \(n\) girls and \(n+4\) boys.

The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\).

REFERENCES:

• Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

INPUT:

• n – positive integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A079923; a
Solutions to the Dancing School problem with n girls and n+4 boys
sage: a.offset
1
sage: a(1)
5
sage: a(8)
15458
sage: a.list(8)
[5, 21, 76, 246, 738, 2108, 5794, 15458]

Python

>>> from sage.all import *
>>> a = sloane.A079923; a
Solutions to the Dancing School problem with n girls and n+4 boys
>>> a.offset
1
>>> a(Integer(1))
5
>>> a(Integer(8))
15458
>>> a.list(Integer(8))
[5, 21, 76, 246, 738, 2108, 5794, 15458]

Compare: Searching Sloane’s online database… Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458]

Sage

sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Python

>>> from sage.all import *
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-17)

class sage.combinat.sloane_functions.A082411

Bases: RecurrenceSequence2

Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

\(a(0) = 407389224418\), \(a(1) = 76343678551\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime, that R. L. Graham in 1964 stated did not contain any primes.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A082411;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
76343678551
sage: a(2)
483732902969
sage: a(3)
560076581520
sage: a(20)
2219759332689173
sage: a.list(4)
[407389224418, 76343678551, 483732902969, 560076581520]

Python

>>> from sage.all import *
>>> a = sloane.A082411;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
>>> a(Integer(1))
76343678551
>>> a(Integer(2))
483732902969
>>> a(Integer(3))
560076581520
>>> a(Integer(20))
2219759332689173
>>> a.list(Integer(4))
[407389224418, 76343678551, 483732902969, 560076581520]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083103

Bases: RecurrenceSequence2

Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

\(a(0) = 1786772701928802632268715130455793\), \(a(1) = 1059683225053915111058165141686995\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co- prime, that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A083103;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
1059683225053915111058165141686995
sage: a(2)
2846455926982717743326880272142788
sage: a(3)
3906139152036632854385045413829783
sage: a.offset
0
sage: a(8)
45481392851206651551714764671352204
sage: a(20)
14639253684254059531823985143948191708
sage: a.list(4)
[1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]

Python

>>> from sage.all import *
>>> a = sloane.A083103;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
>>> a(Integer(1))
1059683225053915111058165141686995
>>> a(Integer(2))
2846455926982717743326880272142788
>>> a(Integer(3))
3906139152036632854385045413829783
>>> a.offset
0
>>> a(Integer(8))
45481392851206651551714764671352204
>>> a(Integer(20))
14639253684254059531823985143948191708
>>> a.list(Integer(4))
[1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083104

Bases: RecurrenceSequence2

Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

\(a(0) = 331635635998274737472200656430763\), \(a(1) = 1510028911088401971189590305498785\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Ronald Graham in 1990.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A083104;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(3)
3351693458175078679851381267428333
sage: a.offset
0
sage: a(8)
36021870400834012982120004949074404
sage: a(20)
11601914177621826012468849361236300628

Python

>>> from sage.all import *
>>> a = sloane.A083104;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
>>> a(Integer(3))
3351693458175078679851381267428333
>>> a.offset
0
>>> a(Integer(8))
36021870400834012982120004949074404
>>> a(Integer(20))
11601914177621826012468849361236300628

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083105

Bases: RecurrenceSequence2

Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

\(a(0) = 62638280004239857\), \(a(1) = 49463435743205655\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Donald Knuth in 1990.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A083105;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
49463435743205655
sage: a(2)
112101715747445512
sage: a(3)
161565151490651167
sage: a.offset
0
sage: a(8)
1853029790662436896
sage: a(20)
596510791500513098192
sage: a.list(4)
[62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]

Python

>>> from sage.all import *
>>> a = sloane.A083105;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
>>> a(Integer(1))
49463435743205655
>>> a(Integer(2))
112101715747445512
>>> a(Integer(3))
161565151490651167
>>> a.offset
0
>>> a(Integer(8))
1853029790662436896
>>> a(Integer(20))
596510791500513098192
>>> a.list(Integer(4))
[62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A083216

Bases: RecurrenceSequence2

Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

\(a(0) = 20615674205555510\), \(a(1) = 3794765361567513\). This is a second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime that does not contain any primes. It was found by Herbert Wilf in 1990.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A083216; a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(0)
20615674205555510
sage: a(1)
3794765361567513
sage: a(8)
347693837265139403
sage: a(41)
2738025383211084205003383
sage: a.list(4)
[20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

Python

>>> from sage.all import *
>>> a = sloane.A083216; a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
>>> a(Integer(0))
20615674205555510
>>> a(Integer(1))
3794765361567513
>>> a(Integer(8))
347693837265139403
>>> a(Integer(41))
2738025383211084205003383
>>> a.list(Integer(4))
[20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

AUTHORS:

• Jaap Spies (2007-01-19)

class sage.combinat.sloane_functions.A090010

Bases: ExtremesOfPermanentsSequence2

Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n\) zeros not on a line.

` a(n) = (n+5)*a(n-1) + (n-1)*a(n-2), a(1)=6, a(2)=43`.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A090010;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
43
sage: a.offset
1
sage: a(8)
67741129
sage: a(22)
192416593029158989003270143
sage: a.list(9)
[6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

Python

>>> from sage.all import *
>>> a = sloane.A090010;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
6
>>> a(Integer(2))
43
>>> a.offset
1
>>> a(Integer(8))
67741129
>>> a(Integer(22))
192416593029158989003270143
>>> a.list(Integer(9))
[6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090012

Bases: SloaneSequence

Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n-1\) zeros not on a line.

\(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2)\), \(a(1)=3\) and \(a(2)=9\)

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A090012;a
Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(2)
9
sage: a.offset
1
sage: a(8)
890901
sage: a(22)
129020386652297208795129
sage: a.list(9)
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

Python

>>> from sage.all import *
>>> a = sloane.A090012;a
Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
3
>>> a(Integer(2))
9
>>> a.offset
1
>>> a(Integer(8))
890901
>>> a(Integer(22))
129020386652297208795129
>>> a.list(Integer(9))
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090013

Bases: SloaneSequence

Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n-1\) zeros not on a line.

\(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=4, a(2)=16]\)

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A090013;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
4
sage: a(2)
16
sage: a.offset
1
sage: a(8)
3481096
sage: a(22)
1112998577171142607670336
sage: a.list(9)
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

Python

>>> from sage.all import *
>>> a = sloane.A090013;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
4
>>> a(Integer(2))
16
>>> a.offset
1
>>> a(Integer(8))
3481096
>>> a(Integer(22))
1112998577171142607670336
>>> a.list(Integer(9))
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090014

Bases: SloaneSequence

Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n-1\) zeros not on a line.

\(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=5, a(2)=25]\)

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A090014;a
Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
5
sage: a(2)
25
sage: a.offset
1
sage: a(8)
11016595
sage: a(22)
7469733600354446865509725
sage: a.list(9)
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

Python

>>> from sage.all import *
>>> a = sloane.A090014;a
Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
5
>>> a(Integer(2))
25
>>> a.offset
1
>>> a(Integer(8))
11016595
>>> a(Integer(22))
7469733600354446865509725
>>> a.list(Integer(9))
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090015

Bases: SloaneSequence

Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n-1\) zeros not on a line.

\(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=6, a(2)=36]\)

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A090015;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
36
sage: a.offset
1
sage: a(8)
29976192
sage: a(22)
41552258517692116794936876
sage: a.list(9)
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

Python

>>> from sage.all import *
>>> a = sloane.A090015;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
6
>>> a(Integer(2))
36
>>> a.offset
1
>>> a(Integer(8))
29976192
>>> a(Integer(22))
41552258517692116794936876
>>> a.list(Integer(9))
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A090016

Bases: SloaneSequence

Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n-1\) zeros not on a line.

\(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=7, a(2)=49]\)

\(A090016 a(n) = A090010(n-1) + A090010(n), a(1)=7\)

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES:

• Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A090016;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
7
sage: a(2)
49
sage: a.offset
1
sage: a(8)
72737161
sage: a(22)
199341969448774341802426289
sage: a.list(9)
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

Python

>>> from sage.all import *
>>> a = sloane.A090016;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(1))
7
>>> a(Integer(2))
49
>>> a.offset
1
>>> a(Integer(8))
72737161
>>> a(Integer(22))
199341969448774341802426289
>>> a.list(Integer(9))
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

AUTHORS:

• Jaap Spies (2007-01-23)

class sage.combinat.sloane_functions.A109814

Bases: SloaneSequence

The \(n\) th term of the sequence \(a(n)\) is the largest \(k\) such that \(n\) can be written as sum of \(k\) consecutive integers.

By definition, \(n\) is the sum of at most \(a(n)\) consecutive positive integers. Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor \(d\) of \(n\) there is a unique corresponding \(k = min(d,2n/d)\). \(a(n)\) can be alternatively defined as the largest among those \(k\) .

See also

• Wikipedia article Polite_number

• An exercise sheet (with answers) about sums of consecutive integers.

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A109814; a
a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a.list(9)
[1, 1, 2, 1, 2, 3, 2, 1, 3]

Python

>>> from sage.all import *
>>> a = sloane.A109814; a
a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(2))
1
>>> a.list(Integer(9))
[1, 1, 2, 1, 2, 3, 2, 1, 3]

AUTHORS:

• Jaap Spies (2007-01-13)

class sage.combinat.sloane_functions.A111774

Bases: SloaneSequence

Sequence of numbers of the third kind, i.e., numbers that can be written as a sum of at least three consecutive positive integers.

Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A111774; a
Numbers that can be written as a sum of at least three consecutive positive integers.
sage: a(1)
6
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
141
sage: a(156)
209
sage: a(302)
386
sage: a.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a = sloane.A111774; a
Numbers that can be written as a sum of at least three consecutive positive integers.
>>> a(Integer(1))
6
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(100))
141
>>> a(Integer(156))
209
>>> a(Integer(302))
386
>>> a.list(Integer(12))
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHORS:

• Jaap Spies (2007-01-13)

is_number_of_the_third_kind(n)

Return True if and only if \(n\) is a number of the third kind.

A number is of the third kind if it can be written as a sum of at least three consecutive positive integers. Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

• \(n\) – positive integer

OUTPUT:

True if \(n\) is not prime and not a power of 2

EXAMPLES:

Sage

sage: a = sloane.A111774
sage: a.is_number_of_the_third_kind(6)
True
sage: a.is_number_of_the_third_kind(100)
True
sage: a.is_number_of_the_third_kind(16)
False
sage: a.is_number_of_the_third_kind(97)
False

Python

>>> from sage.all import *
>>> a = sloane.A111774
>>> a.is_number_of_the_third_kind(Integer(6))
True
>>> a.is_number_of_the_third_kind(Integer(100))
True
>>> a.is_number_of_the_third_kind(Integer(16))
False
>>> a.is_number_of_the_third_kind(Integer(97))
False

AUTHORS:

• Jaap Spies (2006-12-09)

list(n)

EXAMPLES:

Sage

sage: sloane.A111774.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]

Python

>>> from sage.all import *
>>> sloane.A111774.list(Integer(12))
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]

class sage.combinat.sloane_functions.A111775

Bases: SloaneSequence

Number of ways \(n\) can be written as a sum of at least three consecutive integers.

Powers of 2 and (odd) primes can not be written as a sum of at least three consecutive integers. \(a(n)\) strongly depends on the number of odd divisors of \(n\) (A001227): Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor of \(n\) there is a unique corresponding \(k\), \(k=1\) and \(k=2\) must be excluded.

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A111775; a
Number of ways n can be written as a sum of at least three consecutive integers.

Python

>>> from sage.all import *
>>> a = sloane.A111775; a
Number of ways n can be written as a sum of at least three consecutive integers.

Sage

sage: a(1)
0
sage: a(0)
0

Python

>>> from sage.all import *
>>> a(Integer(1))
0
>>> a(Integer(0))
0

We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.

Sage

sage: a(15)
2

Python

>>> from sage.all import *
>>> a(Integer(15))
2

Sage

sage: a(100)
2
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

Python

>>> from sage.all import *
>>> a(Integer(100))
2
>>> a(Integer(256))
0
>>> a(Integer(29))
0
>>> a.list(Integer(20))
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
>>> a(Integer(1)/Integer(3))
Traceback (most recent call last):
...
TypeError: input must be an int or Integer

AUTHORS:

• Jaap Spies (2006-12-09)

class sage.combinat.sloane_functions.A111787

Bases: SloaneSequence

This function returns the \(n\)-th number of Sloane’s sequence A111787

\(a(n)=0\) if \(n\) is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Let \(p\) be the smallest odd prime divisor of \(n\) then \(a(n) = min(p,2n/p)\).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

INPUT:

• n – non negative integer

OUTPUT:

• integer – function value

EXAMPLES:

Sage

sage: a = sloane.A111787; a
a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
sage: a.offset
1
sage: a(1)
0
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
5
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Python

>>> from sage.all import *
>>> a = sloane.A111787; a
a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
>>> a.offset
1
>>> a(Integer(1))
0
>>> a(Integer(0))
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
>>> a(Integer(100))
5
>>> a(Integer(256))
0
>>> a(Integer(29))
0
>>> a.list(Integer(20))
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
>>> a(-Integer(1))
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

AUTHORS:

• Jaap Spies (2007-01-14)

class sage.combinat.sloane_functions.ExponentialNumbers(a)

Bases: SloaneSequence

A sequence of Exponential numbers.

EXAMPLES:

Sage

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0

Python

>>> from sage.all import *
>>> from sage.combinat.sloane_functions import ExponentialNumbers
>>> ExponentialNumbers(Integer(0))
Sequence of Exponential numbers around 0

class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)

Bases: SloaneSequence

gen(a0, a1, d)

EXAMPLES:

Sage

sage: it = sloane.A000153.gen(0,1,2)
sage: [next(it) for i in range(5)]
[0, 1, 2, 7, 32]

Python

>>> from sage.all import *
>>> it = sloane.A000153.gen(Integer(0),Integer(1),Integer(2))
>>> [next(it) for i in range(Integer(5))]
[0, 1, 2, 7, 32]

list(n)

EXAMPLES:

Sage

sage: sloane.A000153.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]

Python

>>> from sage.all import *
>>> sloane.A000153.list(Integer(8))
[0, 1, 2, 7, 32, 181, 1214, 9403]

class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2(offset=1)

Bases: ExtremesOfPermanentsSequence

gen(a0, a1, d)

EXAMPLES:

Sage

sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
sage: e = ExtremesOfPermanentsSequence2()
sage: it = e.gen(6,43,6)
sage: [next(it) for i in range(5)]
[6, 43, 307, 2542, 23799]

Python

>>> from sage.all import *
>>> from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
>>> e = ExtremesOfPermanentsSequence2()
>>> it = e.gen(Integer(6),Integer(43),Integer(6))
>>> [next(it) for i in range(Integer(5))]
[6, 43, 307, 2542, 23799]

class sage.combinat.sloane_functions.RecurrenceSequence(offset=1)

Bases: SloaneSequence

list(n)

EXAMPLES:

Sage

sage: sloane.A001110.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

Python

>>> from sage.all import *
>>> sloane.A001110.list(Integer(8))
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

class sage.combinat.sloane_functions.RecurrenceSequence2(offset=1)

Bases: SloaneSequence

list(n)

EXAMPLES:

Sage

sage: sloane.A001906.list(10)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

Python

>>> from sage.all import *
>>> sloane.A001906.list(Integer(10))
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

class sage.combinat.sloane_functions.Sloane

Bases: SageObject

A collection of Sloane generating functions.

This class inspects sage.combinat.sloane_functions, accumulating all the SloaneSequence classes starting with ‘A’. These are listed for tab completion, but not instantiated until requested.

EXAMPLES:

Ensure we have lots of entries:

Sage

sage: len(sloane.__dir__()) > 100
True

Python

>>> from sage.all import *
>>> len(sloane.__dir__()) > Integer(100)
True

Ensure none are being incorrectly returned:

Sage

sage: [ None for n in sloane.__dir__() if not n.startswith('A') ]
[]

Python

>>> from sage.all import *
>>> [ None for n in sloane.__dir__() if not n.startswith('A') ]
[]

Ensure we can access dynamic constructions and cache correctly:

Sage

sage: s = sloane.A000587
sage: s is sloane.A000587
True

Python

>>> from sage.all import *
>>> s = sloane.A000587
>>> s is sloane.A000587
True

Ensure that we can access other functions in parent classes:

Sage

sage: sloane.__class__
<class 'sage.combinat.sloane_functions.Sloane'>

Python

>>> from sage.all import *
>>> sloane.__class__
<class 'sage.combinat.sloane_functions.Sloane'>

AUTHORS:

• Nick Alexander

class sage.combinat.sloane_functions.SloaneSequence(offset=1)

Bases: SageObject

Base class for a Sloane integer sequence.

list(n)

Return n terms of the sequence:

sequence[offset], sequence[offset+1], ..., sequence[offset+n-1].

EXAMPLES:

Sage

sage: sloane.A000012.list(4)
[1, 1, 1, 1]

Python

>>> from sage.all import *
>>> sloane.A000012.list(Integer(4))
[1, 1, 1, 1]

sage.combinat.sloane_functions.perm_mh(m, h)

This functions calculates \(f(g,h)\) from Sloane’s sequences A079908-A079928

INPUT:

• m – positive integer

• h – non negative integer

OUTPUT: permanent of the \(m \times (m+h)\) matrix, etc.

EXAMPLES:

Sage

sage: from sage.combinat.sloane_functions import perm_mh
sage: perm_mh(3,3)
36
sage: perm_mh(3,4)
76

Python

>>> from sage.all import *
>>> from sage.combinat.sloane_functions import perm_mh
>>> perm_mh(Integer(3),Integer(3))
36
>>> perm_mh(Integer(3),Integer(4))
76

AUTHORS:

• Jaap Spies (2006)

sage.combinat.sloane_functions.recur_gen2(a0, a1, a2, a3)

homogeneous general second-order linear recurrence generator with fixed coefficients

a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2)

EXAMPLES:

Sage

sage: from sage.combinat.sloane_functions import recur_gen2
sage: it = recur_gen2(1,1,1,1)
sage: [next(it) for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

Python

>>> from sage.all import *
>>> from sage.combinat.sloane_functions import recur_gen2
>>> it = recur_gen2(Integer(1),Integer(1),Integer(1),Integer(1))
>>> [next(it) for i in range(Integer(10))]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

sage.combinat.sloane_functions.recur_gen2b(a0, a1, a2, a3, b)

Inhomogeneous second-order linear recurrence generator with fixed coefficients and \(b = f(n)\)

\(a(0) = a0\), \(a(1) = a1\), \(a(n) = a2*a(n-1) + a3*a(n-2) +f(n)\).

EXAMPLES:

Sage

sage: from sage.combinat.sloane_functions import recur_gen2b
sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
sage: [next(it) for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

Python

>>> from sage.all import *
>>> from sage.combinat.sloane_functions import recur_gen2b
>>> it = recur_gen2b(Integer(1),Integer(1),Integer(1),Integer(1), lambda n: Integer(0))
>>> [next(it) for i in range(Integer(10))]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

sage.combinat.sloane_functions.recur_gen3(a0, a1, a2, a3, a4, a5)

homogeneous general third-order linear recurrence generator with fixed coefficients

a(0) = a0, a(1) = a1, a(2) = a2, a(n) = a3*a(n-1) + a4*a(n-2) + a5*a(n-3)

EXAMPLES:

Sage

sage: from sage.combinat.sloane_functions import recur_gen3
sage: it = recur_gen3(1,1,1,1,1,1)
sage: [next(it) for i in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

Python

>>> from sage.all import *
>>> from sage.combinat.sloane_functions import recur_gen3
>>> it = recur_gen3(Integer(1),Integer(1),Integer(1),Integer(1),Integer(1),Integer(1))
>>> [next(it) for i in range(Integer(10))]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]