The On-Line Encyclopedia of Integer Sequences (OEIS)¶

You can query the OEIS (Online Database of Integer Sequences) through Sage in order to:

identify a sequence from its first terms.
obtain more terms, formulae, references, etc. for a given sequence.

EXAMPLES:

Sage

sage: oeis
The On-Line Encyclopedia of Integer Sequences (https://oeis.org/)

Python

>>> from sage.all import *
>>> oeis
The On-Line Encyclopedia of Integer Sequences (https://oeis.org/)

What about a sequence starting with \(3, 7, 15, 1\) ?

Sage

sage: # optional - internet
sage: search = oeis([3, 7, 15, 1], max_results=4); search   # random
0: A001203: Simple continued fraction expansion of Pi.
1: A240698: Partial sums of divisors of n, cf. A027750.
2: A082495: a(n) = (2^n - 1) mod n.
3: A165416: Irregular array read by rows: The n-th row contains those
            distinct positive integers that each, when written in binary,
            occurs as a substring in binary n.
sage: [u.id() for u in search]                              # random
['A001203', 'A240698', 'A082495', 'A165416']
sage: c = search[0]; c
A001203: Simple continued fraction expansion of Pi.

Python

>>> from sage.all import *
>>> # optional - internet
>>> search = oeis([Integer(3), Integer(7), Integer(15), Integer(1)], max_results=Integer(4)); search   # random
0: A001203: Simple continued fraction expansion of Pi.
1: A240698: Partial sums of divisors of n, cf. A027750.
2: A082495: a(n) = (2^n - 1) mod n.
3: A165416: Irregular array read by rows: The n-th row contains those
            distinct positive integers that each, when written in binary,
            occurs as a substring in binary n.
>>> [u.id() for u in search]                              # random
['A001203', 'A240698', 'A082495', 'A165416']
>>> c = search[Integer(0)]; c
A001203: Simple continued fraction expansion of Pi.

Sage

sage: # optional - internet
sage: c.first_terms(15)
(3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1)
sage: c.examples()
0: Pi = 3.1415926535897932384...
1:    = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
2:    = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...].
sage: c.comments()
0: The first 5821569425 terms were computed by _Eric W. Weisstein_ on Sep 18 2011.
1: The first 10672905501 terms were computed by _Eric W. Weisstein_ on Jul 17 2013.
2: The first 15000000000 terms were computed by _Eric W. Weisstein_ on Jul 27 2013.
3: The first 30113021586 terms were computed by _Syed Fahad_ on Apr 27 2021.

Python

>>> from sage.all import *
>>> # optional - internet
>>> c.first_terms(Integer(15))
(3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1)
>>> c.examples()
0: Pi = 3.1415926535897932384...
1:    = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
2:    = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...].
>>> c.comments()
0: The first 5821569425 terms were computed by _Eric W. Weisstein_ on Sep 18 2011.
1: The first 10672905501 terms were computed by _Eric W. Weisstein_ on Jul 17 2013.
2: The first 15000000000 terms were computed by _Eric W. Weisstein_ on Jul 27 2013.
3: The first 30113021586 terms were computed by _Syed Fahad_ on Apr 27 2021.

Sage

sage: # optional - internet
sage: x = c.natural_object(); type(x)
<class 'sage.rings.continued_fraction.ContinuedFraction_periodic'>
sage: x.convergents()[:7]
[3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317]
sage: RR(x.value())
3.14159265358979
sage: RR(x.value()) == RR(pi)
True

Python

>>> from sage.all import *
>>> # optional - internet
>>> x = c.natural_object(); type(x)
<class 'sage.rings.continued_fraction.ContinuedFraction_periodic'>
>>> x.convergents()[:Integer(7)]
[3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317]
>>> RR(x.value())
3.14159265358979
>>> RR(x.value()) == RR(pi)
True

What about posets? Are they hard to count? To which other structures are they related?

Sage

sage: # optional - internet
sage: [Posets(i).cardinality() for i in range(10)]
[1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231]
sage: oeis(_)
0: A000112: Number of partially ordered sets ("posets") with n unlabeled elements.
sage: p = _[0]
sage: 'hard' in p.keywords()
True
sage: len(p.formulas())
0
sage: len(p.first_terms())
17
sage: p.cross_references(fetch=True)    # random
0: A000798: Number of different quasi-orders (or topologies, or transitive digraphs)
            with n labeled elements.
1: A001035: Number of partially ordered sets ("posets") with n labeled elements
            (or labeled acyclic transitive digraphs).
2: A001930: Number of topologies, or transitive digraphs with n unlabeled nodes.
3: A006057: Number of topologies on n labeled points satisfying axioms T_0-T_4.
4: A079263: Number of constrained mixed models with n factors.
5: A079265: Number of antisymmetric transitive binary relations on n unlabeled points.
6: A263859: Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of posets
            with n elements and rank k (or depth k+1).
7: A316978: Number of factorizations of n into factors > 1 with no equivalent primes.
8: A319559: Number of non-isomorphic T_0 set systems of weight n.
9: A326939: Number of T_0 sets of subsets of {1..n} that cover all n vertices.
10: A326943: Number of T_0 sets of subsets of {1..n} that cover all n vertices and
             are closed under intersection.
...

Python

>>> from sage.all import *
>>> # optional - internet
>>> [Posets(i).cardinality() for i in range(Integer(10))]
[1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231]
>>> oeis(_)
0: A000112: Number of partially ordered sets ("posets") with n unlabeled elements.
>>> p = _[Integer(0)]
>>> 'hard' in p.keywords()
True
>>> len(p.formulas())
0
>>> len(p.first_terms())
17
>>> p.cross_references(fetch=True)    # random
0: A000798: Number of different quasi-orders (or topologies, or transitive digraphs)
            with n labeled elements.
1: A001035: Number of partially ordered sets ("posets") with n labeled elements
            (or labeled acyclic transitive digraphs).
2: A001930: Number of topologies, or transitive digraphs with n unlabeled nodes.
3: A006057: Number of topologies on n labeled points satisfying axioms T_0-T_4.
4: A079263: Number of constrained mixed models with n factors.
5: A079265: Number of antisymmetric transitive binary relations on n unlabeled points.
6: A263859: Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of posets
            with n elements and rank k (or depth k+1).
7: A316978: Number of factorizations of n into factors > 1 with no equivalent primes.
8: A319559: Number of non-isomorphic T_0 set systems of weight n.
9: A326939: Number of T_0 sets of subsets of {1..n} that cover all n vertices.
10: A326943: Number of T_0 sets of subsets of {1..n} that cover all n vertices and
             are closed under intersection.
...

What does the Taylor expansion of the \(e^{e^x-1}\) function have to do with primes ?

Sage

sage: # optional - internet, needs sage.symbolic
sage: x = var('x') ; f(x) = e^(e^x - 1)
sage: L = [a*factorial(b) for a,b in taylor(f(x), x, 0, 20).coefficients()]; L
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597,
 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159,
 5832742205057, 51724158235372]
sage: oeis(L)
0: A000110: Bell or exponential numbers: number of ways to partition
            a set of n labeled elements.
1: A292935: E.g.f.: exp(exp(-x) - 1).
sage: b = _[0]
sage: b.formulas()[0]
'E.g.f.: exp(exp(x) - 1).'
sage: [i for i in b.comments() if 'prime' in i][-1]
'When n is prime, ...'
sage: [n for n in range(2, 20) if (b(n)-2) % n == 0]
[2, 3, 5, 7, 11, 13, 17, 19]

Python

>>> from sage.all import *
>>> # optional - internet, needs sage.symbolic
>>> x = var('x') ; __tmp__=var("x"); f = symbolic_expression(e**(e**x - Integer(1))).function(x)
>>> L = [a*factorial(b) for a,b in taylor(f(x), x, Integer(0), Integer(20)).coefficients()]; L
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597,
 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159,
 5832742205057, 51724158235372]
>>> oeis(L)
0: A000110: Bell or exponential numbers: number of ways to partition
            a set of n labeled elements.
1: A292935: E.g.f.: exp(exp(-x) - 1).
>>> b = _[Integer(0)]
>>> b.formulas()[Integer(0)]
'E.g.f.: exp(exp(x) - 1).'
>>> [i for i in b.comments() if 'prime' in i][-Integer(1)]
'When n is prime, ...'
>>> [n for n in range(Integer(2), Integer(20)) if (b(n)-Integer(2)) % n == Integer(0)]
[2, 3, 5, 7, 11, 13, 17, 19]

See also

If you plan to do a lot of automatic searches for subsequences, you should consider installing SloaneEncyclopedia, a local partial copy of the OEIS.
Some infinite OEIS sequences are implemented in Sage, via the sloane_functions module.

AUTHORS:

Thierry Monteil (2012-02-10 – 2013-06-21): initial version.
Vincent Delecroix (2014): modifies continued fractions because of Issue #14567
Moritz Firsching (2016): modifies handling of dead sequence, see Issue #17330
Thierry Monteil (2019): refactorization (unique representation Issue #28480, laziness Issue #28627)

class sage.databases.oeis.FancyTuple(iterable=(), /)[source]¶

Bases: tuple

This class inherits from tuple, it allows to nicely print tuples whose elements have a one line representation.

EXAMPLES:

Sage

sage: from sage.databases.oeis import FancyTuple
sage: t = FancyTuple(['zero', 'one', 'two', 'three', 4]); t
0: zero
1: one
2: two
3: three
4: 4

sage: t[2]
'two'

Python

>>> from sage.all import *
>>> from sage.databases.oeis import FancyTuple
>>> t = FancyTuple(['zero', 'one', 'two', 'three', Integer(4)]); t
0: zero
1: one
2: two
3: three
4: 4

>>> t[Integer(2)]
'two'

class sage.databases.oeis.OEIS[source]¶

Bases: object

The On-Line Encyclopedia of Integer Sequences.

OEIS is a class representing the On-Line Encyclopedia of Integer Sequences. You can query it using its methods, but OEIS can also be called directly with three arguments:

query – it can be:
- a string representing an OEIS ID (e.g. ‘A000045’).
- an integer representing an OEIS ID (e.g. 45).
- a list representing a sequence of integers.
- a string, representing a text search.
max_results – (integer, default: 30) the maximum number of results to return, they are sorted according to their relevance. In any cases, the OEIS website will never provide more than 100 results.
first_result – (integer, default: 0) allow to skip the first_result first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences.

OUTPUT:

if query is an integer or an OEIS ID (e.g. ‘A000045’), returns the associated OEIS sequence.
if query is a string, returns a tuple of OEIS sequences whose description corresponds to the query. Those sequences can be used without the need to fetch the database again.
if query is a list or tuple of integers, returns a tuple of OEIS sequences containing it as a subsequence. Those sequences can be used without the need to fetch the database again.

EXAMPLES:

Sage

sage: oeis
The On-Line Encyclopedia of Integer Sequences (https://oeis.org/)

Python

>>> from sage.all import *
>>> oeis
The On-Line Encyclopedia of Integer Sequences (https://oeis.org/)

A particular sequence can be called by its A-number or number:

Sage

sage: oeis('A000040')                           # optional -- internet
A000040: The prime numbers.
sage: oeis(45)                                  # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

Python

>>> from sage.all import *
>>> oeis('A000040')                           # optional -- internet
A000040: The prime numbers.
>>> oeis(Integer(45))                                  # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

The database can be searched by subsequence:

Sage

sage: search = oeis([1,2,3,5,8,13]); search     # optional -- internet
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
1: A290689: Number of transitive rooted trees with n nodes.
2: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0;
            T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1)
            for k = 2..2n-1, n >= 2.

sage: fibo = search[0]                          # optional -- internet

sage: fibo.name()                               # optional -- internet
'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.'

sage: print(fibo.first_terms())                 # optional -- internet
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,
2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418,
317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465,
14930352, 24157817, 39088169, 63245986, 102334155)

sage: fibo.cross_references()[0]                # optional -- internet
'A001622'

sage: fibo == oeis(45)                          # optional -- internet
True

sage: sfibo = oeis('A039834')                   # optional -- internet
sage: sfibo.first_terms()                       # optional -- internet
(1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, 233,
-377, 610, -987, 1597, -2584, 4181, -6765, 10946, -17711, 28657,
-46368, 75025, -121393, 196418, -317811, 514229, -832040, 1346269,
-2178309, 3524578, -5702887, 9227465, -14930352, 24157817)

sage: tuple(abs(i) for i in sfibo.first_terms())[2:20] == fibo.first_terms()[:18]   # optional -- internet
True

sage: fibo.formulas()[4]                        # optional -- internet
'F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).'

sage: fibo.comments()[6]                        # optional -- internet
"F(n+2) = number of binary sequences of length n that have no
consecutive 0's."

sage: fibo.links()[0]                           # optional -- internet
'https://oeis.org/A000045/b000045.txt'

Python

>>> from sage.all import *
>>> search = oeis([Integer(1),Integer(2),Integer(3),Integer(5),Integer(8),Integer(13)]); search     # optional -- internet
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
1: A290689: Number of transitive rooted trees with n nodes.
2: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0;
            T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1)
            for k = 2..2n-1, n >= 2.

>>> fibo = search[Integer(0)]                          # optional -- internet

>>> fibo.name()                               # optional -- internet
'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.'

>>> print(fibo.first_terms())                 # optional -- internet
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,
2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418,
317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465,
14930352, 24157817, 39088169, 63245986, 102334155)

>>> fibo.cross_references()[Integer(0)]                # optional -- internet
'A001622'

>>> fibo == oeis(Integer(45))                          # optional -- internet
True

>>> sfibo = oeis('A039834')                   # optional -- internet
>>> sfibo.first_terms()                       # optional -- internet
(1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, 233,
-377, 610, -987, 1597, -2584, 4181, -6765, 10946, -17711, 28657,
-46368, 75025, -121393, 196418, -317811, 514229, -832040, 1346269,
-2178309, 3524578, -5702887, 9227465, -14930352, 24157817)

>>> tuple(abs(i) for i in sfibo.first_terms())[Integer(2):Integer(20)] == fibo.first_terms()[:Integer(18)]   # optional -- internet
True

>>> fibo.formulas()[Integer(4)]                        # optional -- internet
'F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).'

>>> fibo.comments()[Integer(6)]                        # optional -- internet
"F(n+2) = number of binary sequences of length n that have no
consecutive 0's."

>>> fibo.links()[Integer(0)]                           # optional -- internet
'https://oeis.org/A000045/b000045.txt'

The database can be searched by description:

Sage

sage: oeis('prime gap factorization', max_results=4)  # random  # optional -- internet
0: A073491: Numbers having no prime gaps in their factorization.
1: A073485: Product of any number of consecutive primes; squarefree numbers
            with no gaps in their prime factorization.
2: A073490: Number of prime gaps in factorization of n.
3: A073492: Numbers having at least one prime gap in their factorization.

Python

>>> from sage.all import *
>>> oeis('prime gap factorization', max_results=Integer(4))  # random  # optional -- internet
0: A073491: Numbers having no prime gaps in their factorization.
1: A073485: Product of any number of consecutive primes; squarefree numbers
            with no gaps in their prime factorization.
2: A073490: Number of prime gaps in factorization of n.
3: A073492: Numbers having at least one prime gap in their factorization.

Note

The following will fetch the OEIS database only once:

Sage

sage: oeis([1,2,3,5,8,13])                  # optional -- internet
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
1: A290689: Number of transitive rooted trees with n nodes.
2: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0;
            T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1)
            for k = 2..2n-1, n >= 2.

sage: oeis('A000045')                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

Python

>>> from sage.all import *
>>> oeis([Integer(1),Integer(2),Integer(3),Integer(5),Integer(8),Integer(13)])                  # optional -- internet
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
1: A290689: Number of transitive rooted trees with n nodes.
2: A027926: Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0;
            T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1)
            for k = 2..2n-1, n >= 2.

>>> oeis('A000045')                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

Indeed, due to some caching mechanism, the sequence is not re-created when called from its ID.

browse()[source]¶

Open the OEIS web page in a browser.

EXAMPLES:

Sage

sage: oeis.browse()                         # optional -- webbrowser

Python

>>> from sage.all import *
>>> oeis.browse()                         # optional -- webbrowser

find_by_description(description, max_results=3, first_result=0)[source]¶

Search for OEIS sequences corresponding to the description.

INPUT:

description – string; the description the searched sequences
max_results – (integer, default: 3) the maximum number of results we want. In any case, the on-line encyclopedia will not return more than 100 results.
first_result – (integer, default: 0) allow to skip the first_result first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences.

OUTPUT:

a tuple (with fancy formatting) of at most max_results OEIS sequences. Those sequences can be used without the need to fetch the database again.

EXAMPLES:

Sage

sage: oeis.find_by_description('prime gap factorization') # optional -- internet
0: A...: ...
1: A...: ...
2: A...: ...

sage: prime_gaps = _[2] ; prime_gaps        # optional -- internet
A...

sage: oeis('beaver')                        # optional -- internet
0: A...: ...eaver...
1: A...: ...eaver...
2: A...: ...eaver...

sage: oeis('beaver', max_results=4, first_result=2)     # optional -- internet
0: A...: ...eaver...
1: A...: ...eaver...
2: A...: ...eaver...
3: A...: ...eaver...

Python

>>> from sage.all import *
>>> oeis.find_by_description('prime gap factorization') # optional -- internet
0: A...: ...
1: A...: ...
2: A...: ...

>>> prime_gaps = _[Integer(2)] ; prime_gaps        # optional -- internet
A...

>>> oeis('beaver')                        # optional -- internet
0: A...: ...eaver...
1: A...: ...eaver...
2: A...: ...eaver...

>>> oeis('beaver', max_results=Integer(4), first_result=Integer(2))     # optional -- internet
0: A...: ...eaver...
1: A...: ...eaver...
2: A...: ...eaver...
3: A...: ...eaver...

find_by_entry(entry)[source]¶

INPUT:

entry – string corresponding to an entry in the internal format of the OEIS

OUTPUT: the corresponding OEIS sequence

EXAMPLES:

Sage

sage: entry = '%I A262002\n%N A262002 L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).\n%K A262002 nonn'
sage: s = oeis.find_by_entry(entry)
sage: s
A262002: L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).

Python

>>> from sage.all import *
>>> entry = '%I A262002\n%N A262002 L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).\n%K A262002 nonn'
>>> s = oeis.find_by_entry(entry)
>>> s
A262002: L.g.f.: log( Sum_{n>=0} x^n/n! * Product_{k=1..n} (k^2 + 1) ).

find_by_id(ident, fetch=False)[source]¶

INPUT:

ident – string representing the A-number of the sequence or an integer representing its number
fetch – boolean (default: False); whether to force fetching the content of the sequence on the internet

OUTPUT: the OEIS sequence whose A-number or number corresponds to ident

EXAMPLES:

Sage

sage: oeis.find_by_id('A000040')            # optional -- internet
A000040: The prime numbers.

sage: oeis.find_by_id(40)                   # optional -- internet
A000040: The prime numbers.

Python

>>> from sage.all import *
>>> oeis.find_by_id('A000040')            # optional -- internet
A000040: The prime numbers.

>>> oeis.find_by_id(Integer(40))                   # optional -- internet
A000040: The prime numbers.

find_by_subsequence(subsequence, max_results=3, first_result=0)[source]¶

Search for OEIS sequences containing the given subsequence.

INPUT:

subsequence – list or tuple of integers
max_results – integer (default: 3); the maximum of results requested
first_result – integer (default: 0); allow to skip the first_result first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences.

OUTPUT:

A tuple (with fancy formatting) of at most max_results OEIS sequences. Those sequences can be used without the need to fetch the database again.

EXAMPLES:

Sage

sage: oeis.find_by_subsequence([2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377])  # optional -- internet # random
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
1: A212804: Expansion of (1 - x)/(1 - x - x^2).
2: A020695: Pisot sequence E(2,3).

sage: fibo = _[0] ; fibo                    # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

Python

>>> from sage.all import *
>>> oeis.find_by_subsequence([Integer(2), Integer(3), Integer(5), Integer(8), Integer(13), Integer(21), Integer(34), Integer(55), Integer(89), Integer(144), Integer(233), Integer(377)])  # optional -- internet # random
0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
1: A212804: Expansion of (1 - x)/(1 - x - x^2).
2: A020695: Pisot sequence E(2,3).

>>> fibo = _[Integer(0)] ; fibo                    # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

class sage.databases.oeis.OEISSequence(ident)[source]¶

Bases: SageObject, UniqueRepresentation

The class of OEIS sequences.

This class implements OEIS sequences. They are usually produced by calls to the On-Line Encyclopedia of Integer Sequences, represented by the class OEIS.

Note

Since some sequences do not start with index 0, there is a difference between calling and getting item, see __call__() for more details

Sage

sage: # optional - internet
sage: sfibo = oeis('A039834')
sage: sfibo.first_terms()[:10]
(1, 1, 0, 1, -1, 2, -3, 5, -8, 13)
sage: sfibo(-2)
1
sage: sfibo(3)
2
sage: sfibo.offsets()
(-2, 6)
sage: sfibo[0]
1
sage: sfibo[6]
-3

Python

>>> from sage.all import *
>>> # optional - internet
>>> sfibo = oeis('A039834')
>>> sfibo.first_terms()[:Integer(10)]
(1, 1, 0, 1, -1, 2, -3, 5, -8, 13)
>>> sfibo(-Integer(2))
1
>>> sfibo(Integer(3))
2
>>> sfibo.offsets()
(-2, 6)
>>> sfibo[Integer(0)]
1
>>> sfibo[Integer(6)]
-3

__call__(k)[source]¶

Return the element of the sequence self with index k.

INPUT:

k – integer

OUTPUT: integer

Note

The first index of the sequence self is not necessarily zero, it depends on the first offset of self. If the sequence represents the decimal expansion of a real number, the index 0 corresponds to the digit right after the decimal point.

EXAMPLES:

Sage

sage: f = oeis(45)                          # optional -- internet
sage: f.first_terms()[:10]                  # optional -- internet
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34)

sage: f(4)                                  # optional -- internet
3

Python

>>> from sage.all import *
>>> f = oeis(Integer(45))                          # optional -- internet
>>> f.first_terms()[:Integer(10)]                  # optional -- internet
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34)

>>> f(Integer(4))                                  # optional -- internet
3

Sage

sage: sfibo = oeis('A039834')               # optional -- internet
sage: sfibo.first_terms()[:10]              # optional -- internet
(1, 1, 0, 1, -1, 2, -3, 5, -8, 13)

sage: sfibo(-2)                             # optional -- internet
1
sage: sfibo(4)                              # optional -- internet
-3
sage: sfibo.offsets()                       # optional -- internet
(-2, 6)

Python

>>> from sage.all import *
>>> sfibo = oeis('A039834')               # optional -- internet
>>> sfibo.first_terms()[:Integer(10)]              # optional -- internet
(1, 1, 0, 1, -1, 2, -3, 5, -8, 13)

>>> sfibo(-Integer(2))                             # optional -- internet
1
>>> sfibo(Integer(4))                              # optional -- internet
-3
>>> sfibo.offsets()                       # optional -- internet
(-2, 6)

author()[source]¶

Return the author of the sequence in the encyclopedia.

OUTPUT: string

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.author()                            # optional -- internet
'_N. J. A. Sloane_, 1964'

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.author()                            # optional -- internet
'_N. J. A. Sloane_, 1964'

browse()[source]¶

Open the OEIS web page associated to the sequence self in a browser.

EXAMPLES:

Sage

sage: f = oeis(45); f                       # optional -- internet webbrowser
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.browse()                            # optional -- internet webbrowser

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)); f                       # optional -- internet webbrowser
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.browse()                            # optional -- internet webbrowser

comments()[source]¶

Return a tuple of comments associated to the sequence self.

OUTPUT:

tuple of strings (with fancy formatting).

EXAMPLES:

Sage

sage: f = oeis(45); f                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.comments()[:8]                      # optional -- internet
D. E. Knuth writes...
In keeping with historical accounts...
Susantha Goonatilake writes...
Also sometimes called Hemachandra numbers.
Also sometimes called Lamé's sequence.
...
F(n+2) = number of binary sequences of length n that have no consecutive 0's.
F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive integers.

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)); f                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.comments()[:Integer(8)]                      # optional -- internet
0: D. E. Knuth writes...
1: In keeping with historical accounts...
2: Susantha Goonatilake writes...
3: Also sometimes called Hemachandra numbers.
4: Also sometimes called Lamé's sequence.
5: ...
6: F(n+2) = number of binary sequences of length n that have no consecutive 0's.
7: F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive integers.

cross_references(fetch=False)[source]¶

Return a tuple of cross references associated to the sequence self.

INPUT:

fetch – boolean (default: False)

OUTPUT:

if fetch is False, return a list of OEIS IDs (strings).
if fetch is True, return a tuple of OEIS sequences.

EXAMPLES:

Sage

sage: nbalanced = oeis("A005598") ; nbalanced   # optional -- internet
A005598: a(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i).

sage: nbalanced.cross_references()              # optional -- internet
('A000010', 'A002088', 'A011755', 'A049695', 'A049703', 'A103116')

sage: nbalanced.cross_references(fetch=True)    # optional -- internet
0: A000010: Euler totient function phi(n): count numbers <= n and prime to n.
1: A002088: Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.
2: A011755: a(n) = Sum_{k=1..n} k*phi(k).
3: A049695: Array T read by diagonals; T(i,j) is the number of nonnegative
            slopes of lines determined by 2 lattice points in
            [ 0,i ] X [ 0,j ] if i > 0; T(0,j)=1 if j > 0; T(0,0)=0.
4: A049703: a(0) = 0; for n>0, a(n) = A005598(n)/2.
5: A103116: a(n) = Sum_{i=1..n} (n-i+1)*phi(i).

sage: phi = _[3]                                # optional -- internet

Python

>>> from sage.all import *
>>> nbalanced = oeis("A005598") ; nbalanced   # optional -- internet
A005598: a(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i).

>>> nbalanced.cross_references()              # optional -- internet
('A000010', 'A002088', 'A011755', 'A049695', 'A049703', 'A103116')

>>> nbalanced.cross_references(fetch=True)    # optional -- internet
0: A000010: Euler totient function phi(n): count numbers <= n and prime to n.
1: A002088: Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010.
2: A011755: a(n) = Sum_{k=1..n} k*phi(k).
3: A049695: Array T read by diagonals; T(i,j) is the number of nonnegative
            slopes of lines determined by 2 lattice points in
            [ 0,i ] X [ 0,j ] if i > 0; T(0,j)=1 if j > 0; T(0,0)=0.
4: A049703: a(0) = 0; for n>0, a(n) = A005598(n)/2.
5: A103116: a(n) = Sum_{i=1..n} (n-i+1)*phi(i).

>>> phi = _[Integer(3)]                                # optional -- internet

examples()[source]¶

Return a tuple of examples associated to the sequence self.

OUTPUT:

tuple of strings (with fancy formatting).

EXAMPLES:

Sage

sage: c = oeis(1203); c                     # optional -- internet
A001203: Simple continued fraction expansion of Pi.

sage: c.examples()                          # optional -- internet
0: Pi = 3.1415926535897932384...
1:    = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
2:    = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...].

Python

>>> from sage.all import *
>>> c = oeis(Integer(1203)); c                     # optional -- internet
A001203: Simple continued fraction expansion of Pi.

>>> c.examples()                          # optional -- internet
0: Pi = 3.1415926535897932384...
1:    = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
2:    = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 1, 292, ...].

extensions_or_errors()[source]¶

Return a tuple of extensions or errors associated to the sequence self.

OUTPUT:

tuple of strings (with fancy formatting).

EXAMPLES:

Sage

sage: sfibo = oeis('A039834'); sfibo        # optional -- internet
A039834: a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with
         a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended
         to negative indices.

sage: sfibo.extensions_or_errors()[0]       # optional -- internet
'Signs corrected by _Len Smiley_ and _N. J. A. Sloane_'

Python

>>> from sage.all import *
>>> sfibo = oeis('A039834'); sfibo        # optional -- internet
A039834: a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with
         a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended
         to negative indices.

>>> sfibo.extensions_or_errors()[Integer(0)]       # optional -- internet
'Signs corrected by _Len Smiley_ and _N. J. A. Sloane_'

first_terms(number=None)[source]¶

INPUT:

number – integer or None (default); the number of terms returned (if less than the number of available terms). When set to None, returns all the known terms.

OUTPUT: tuple of integers

EXAMPLES:

Sage

sage: f = oeis(45); f                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.first_terms()[:10]                  # optional -- internet
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34)

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)); f                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.first_terms()[:Integer(10)]                  # optional -- internet
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34)

Handle dead sequences, see Issue #17330

Sage

sage: oeis(5000).first_terms(12)            # optional -- internet
(1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2537)

Python

>>> from sage.all import *
>>> oeis(Integer(5000)).first_terms(Integer(12))            # optional -- internet
(1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2537)

formulas()[source]¶

Return a tuple of formulas associated to the sequence self.

OUTPUT:

tuple of strings (with fancy formatting).

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.formulas()[2]                       # optional -- internet
'F(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)).'

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.formulas()[Integer(2)]                       # optional -- internet
'F(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)).'

id(format='A')[source]¶

The ID of the sequence self is the A-number that identifies self.

INPUT:

format – string (default: 'A')

OUTPUT:

if format is set to ‘A’, returns a string of the form ‘A000123’.
if format is set to ‘int’ returns an integer of the form 123.

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.id()                                # optional -- internet
'A000045'

sage: f.id(format='int')                    # optional -- internet
45

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.id()                                # optional -- internet
'A000045'

>>> f.id(format='int')                    # optional -- internet
45

is_dead(warn_only=False)[source]¶

Tell whether the sequence is dead.

INPUT:

warn_only – ignored

EXAMPLES:

sage: s = oeis(17) # optional – internet sage: s # optional – internet A000017: Erroneous version of A032522.

is_finite()[source]¶

Tell whether the sequence is finite.

Currently, OEIS only provides a keyword when the sequence is known to be finite. So, when this keyword is not there, we do not know whether it is infinite or not.

OUTPUT:

True when the sequence is known to be finite.
Unknown otherwise.

Todo

Ask OEIS for a keyword ensuring that a sequence is infinite.

EXAMPLES:

Sage

sage: s = oeis('A114288') ; s               # optional -- internet
A114288: Lexicographically earliest solution of any 9 X 9 sudoku, read by rows.

sage: s.is_finite()                         # optional -- internet
True

Python

>>> from sage.all import *
>>> s = oeis('A114288') ; s               # optional -- internet
A114288: Lexicographically earliest solution of any 9 X 9 sudoku, read by rows.

>>> s.is_finite()                         # optional -- internet
True

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.is_finite()                         # optional -- internet
Unknown

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.is_finite()                         # optional -- internet
Unknown

is_full()[source]¶

Tell whether the sequence self is full, that is, if all its elements are listed in self.first_terms().

Currently, OEIS only provides a keyword when the sequence is known to be full. So, when this keyword is not there, we do not know whether some elements are missing or not.

OUTPUT:

True when the sequence is known to be full.
Unknown otherwise.

EXAMPLES:

Sage

sage: s = oeis('A114288') ; s               # optional -- internet
A114288: Lexicographically earliest solution of any 9 X 9 sudoku, read by rows.

sage: s.is_full()                           # optional -- internet
True

Python

>>> from sage.all import *
>>> s = oeis('A114288') ; s               # optional -- internet
A114288: Lexicographically earliest solution of any 9 X 9 sudoku, read by rows.

>>> s.is_full()                           # optional -- internet
True

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.is_full()                           # optional -- internet
Unknown

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.is_full()                           # optional -- internet
Unknown

keywords(warn=True)[source]¶

Return the keywords associated to the sequence self.

OUTPUT: tuple of strings

EXAMPLES:

Sage

sage: f = oeis(53) ; f                      # optional -- internet
A000053: Local stops on New York City...

sage: f.keywords()                          # optional -- internet
('nonn', 'fini', ...)

Python

>>> from sage.all import *
>>> f = oeis(Integer(53)) ; f                      # optional -- internet
A000053: Local stops on New York City...

>>> f.keywords()                          # optional -- internet
('nonn', 'fini', ...)

links(browse=None, format='guess')[source]¶

Return, display or browse links associated to the sequence self.

INPUT:

browse – integer; a list of integers, or the word ‘all’ (default: None) which links to open in a web browser
format – string (default: 'guess'); how to display the links

OUTPUT: tuple of strings (with fancy formatting):

if format is url, returns a tuple of absolute links without description.
if format is html, returns nothing but prints a tuple of clickable absolute links in their context.
if format is guess, adapts the output to the context (command line or notebook).
if format is raw, the links as they appear in the database, relative links are not made absolute.

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.links(format='url')                             # optional -- internet
0: https://oeis.org/A000045/b000045.txt
1: ...
2: ...

sage: f.links(format='raw')                 # optional -- internet
0: N. J. A. Sloane, <a href="/A000045/b000045.txt">The first 2000 Fibonacci numbers: Table of n, F(n) for n = 0..2000</a>
1: ...
2: ...

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.links(format='url')                             # optional -- internet
0: https://oeis.org/A000045/b000045.txt
1: ...
2: ...

>>> f.links(format='raw')                 # optional -- internet
0: N. J. A. Sloane, <a href="/A000045/b000045.txt">The first 2000 Fibonacci numbers: Table of n, F(n) for n = 0..2000</a>
1: ...
2: ...

name()[source]¶

Return the name of the sequence self.

OUTPUT: string

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.name()                              # optional -- internet
'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.'

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.name()                              # optional -- internet
'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.'

natural_object()[source]¶

Return the natural object associated to the sequence self.

OUTPUT:

If the sequence self corresponds to the digits of a real number, returns the associated real number (as an element of RealLazyField()).
If the sequence self corresponds to the convergents of a continued fraction, returns the associated continued fraction.

Warning

This method forgets the fact that the returned sequence may not be complete.

Todo

ask OEIS to add a keyword telling whether the sequence comes from a power series, e.g. for OEIS sequence A000182
discover other possible conversions.

EXAMPLES:

Sage

sage: g = oeis("A002852"); g                # optional -- internet
A002852: Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.

sage: x = g.natural_object(); type(x)       # optional -- internet
<class 'sage.rings.continued_fraction.ContinuedFraction_periodic'>

sage: RDF(x) == RDF(euler_gamma)            # optional -- internet
True

sage: cfg = continued_fraction(euler_gamma)                                 # needs sage.symbolic
sage: x[:90] == cfg[:90]                    # optional - internet           # needs sage.symbolic
True

Python

>>> from sage.all import *
>>> g = oeis("A002852"); g                # optional -- internet
A002852: Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.

>>> x = g.natural_object(); type(x)       # optional -- internet
<class 'sage.rings.continued_fraction.ContinuedFraction_periodic'>

>>> RDF(x) == RDF(euler_gamma)            # optional -- internet
True

>>> cfg = continued_fraction(euler_gamma)                                 # needs sage.symbolic
>>> x[:Integer(90)] == cfg[:Integer(90)]                    # optional - internet           # needs sage.symbolic
True

Sage

sage: ee = oeis('A001113'); ee              # optional -- internet
A001113: Decimal expansion of e.

sage: x = ee.natural_object(); x            # optional -- internet
2.718281828459046?

sage: x.parent()                            # optional -- internet
Real Lazy Field

sage: x == RR(e)                            # optional -- internet
True

Python

>>> from sage.all import *
>>> ee = oeis('A001113'); ee              # optional -- internet
A001113: Decimal expansion of e.

>>> x = ee.natural_object(); x            # optional -- internet
2.718281828459046?

>>> x.parent()                            # optional -- internet
Real Lazy Field

>>> x == RR(e)                            # optional -- internet
True

Sage

sage: av = oeis('A322578'); av              # optional -- internet
A322578: Decimal expansion ... Avogadro...

sage: av.natural_object()                   # optional -- internet
6.022140760000000?e23

Python

>>> from sage.all import *
>>> av = oeis('A322578'); av              # optional -- internet
A322578: Decimal expansion ... Avogadro...

>>> av.natural_object()                   # optional -- internet
6.022140760000000?e23

Sage

sage: fib = oeis('A000045'); fib            # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: x = fib.natural_object(); x.universe()    # optional -- internet
Non negative integer semiring

Python

>>> from sage.all import *
>>> fib = oeis('A000045'); fib            # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> x = fib.natural_object(); x.universe()    # optional -- internet
Non negative integer semiring

Sage

sage: sfib = oeis('A039834'); sfib         # optional -- internet
A039834: a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with
         a(-2) = a(-1) = 1; or Fibonacci numbers (A000045)
         extended to negative indices.

sage: x = sfib.natural_object(); x.universe()   # optional -- internet
Integer Ring

Python

>>> from sage.all import *
>>> sfib = oeis('A039834'); sfib         # optional -- internet
A039834: a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with
         a(-2) = a(-1) = 1; or Fibonacci numbers (A000045)
         extended to negative indices.

>>> x = sfib.natural_object(); x.universe()   # optional -- internet
Integer Ring

offsets()[source]¶

Return the offsets of the sequence self.

The first offset is the subscript of the first term in the sequence self. When, the sequence represents the decimal expansion of a real number, it corresponds to the number of digits of its integer part.

The second offset is the first term in the sequence self (starting from 1) whose absolute value is greater than 1. This is set to 1 if all the terms are 0 or +-1.

OUTPUT: tuple of two elements

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.offsets()                           # optional -- internet
(0, 4)

sage: f.first_terms()[:4]                   # optional -- internet
(0, 1, 1, 2)

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.offsets()                           # optional -- internet
(0, 4)

>>> f.first_terms()[:Integer(4)]                   # optional -- internet
(0, 1, 1, 2)

old_IDs()[source]¶

Return the IDs of the sequence self corresponding to ancestors of OEIS.

OUTPUT:

a tuple of at most two strings. When the string starts with \(M\), it corresponds to the ID of “The Encyclopedia of Integer Sequences” of 1995. When the string starts with \(N\), it corresponds to the ID of the “Handbook of Integer Sequences” of 1973.

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.old_IDs()                           # optional -- internet
('M0692', 'N0256')

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.old_IDs()                           # optional -- internet
('M0692', 'N0256')

online_update()[source]¶: Fetch the online OEIS to update the informations about this sequence.

programs(language='all', preparsing=True, keep_comments=False)[source]¶

Return programs for the sequence self in the given language.

INPUT:

language – string (default: 'all'); the chosen language. Possible values are ‘all’ for the full list, or any language name, for example ‘sage’, ‘maple’, ‘mathematica’, etc.

Some further optional input is specific to sage code treatment:

preparsing – boolean (default: True); whether to preparse sage code
keep_comments – boolean (default: False); whether to keep comments in sage code

OUTPUT:

If language is 'all', this returns a sorted list of pairs (language, code), where every language can appear several times.

Otherwise, this returns a list of programs in the language, each program being a tuple of strings (with fancy formatting).

EXAMPLES:

Sage

sage: ee = oeis('A001113') ; ee             # optional -- internet
A001113: Decimal expansion of e.

sage: ee.programs('pari')[0]                # optional -- internet
0: default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 15 2009

sage: G = oeis.find_by_id('A27642')   # optional -- internet
sage: G.programs('all')               # optional -- internet
[('haskell', ...),
 ('magma', ...),
 ...
 ('python', ...),
 ('sage', ...)]

Python

>>> from sage.all import *
>>> ee = oeis('A001113') ; ee             # optional -- internet
A001113: Decimal expansion of e.

>>> ee.programs('pari')[Integer(0)]                # optional -- internet
0: default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 15 2009

>>> G = oeis.find_by_id('A27642')   # optional -- internet
>>> G.programs('all')               # optional -- internet
[('haskell', ...),
 ('magma', ...),
 ...
 ('python', ...),
 ('sage', ...)]

raw_entry()[source]¶

Return the raw entry of the sequence self, in the OEIS format.

The raw entry is fetched online if needed.

OUTPUT: string

EXAMPLES:

Sage

sage: f = oeis(45) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: print(f.raw_entry())                  # optional -- internet
%I A000045 M0692 N0256...
%S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,...
%T A000045 10946,17711,28657,46368,...
...

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)) ; f                      # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> print(f.raw_entry())                  # optional -- internet
%I A000045 M0692 N0256...
%S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,...
%T A000045 10946,17711,28657,46368,...
...

references()[source]¶

Return a tuple of references associated to the sequence self.

OUTPUT:

tuple of strings (with fancy formatting).

EXAMPLES:

Sage

sage: w = oeis(7540) ; w                    # optional -- internet
A007540: Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

sage: w.references()                        # optional -- internet
...Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.
...

Python

>>> from sage.all import *
>>> w = oeis(Integer(7540)) ; w                    # optional -- internet
A007540: Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

>>> w.references()                        # optional -- internet
...Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.
...

show()[source]¶

Display most available information about the sequence self.

EXAMPLES:

Sage

sage: s = oeis(12345)                       # optional -- internet
sage: s.show()                              # optional -- internet
ID
A012345

NAME
Coefficients in the expansion sinh(arcsin(x)*arcsin(x)) = 2*x^2/2!+8*x^4/4!+248*x^6/6!+11328*x^8/8!+...

FIRST TERMS
(2, 8, 248, 11328, 849312, 94857600, 14819214720, 3091936512000, 831657655349760, 280473756197529600, 115967597965430077440, 57712257892456911912960, 34039765801079493369569280)

LINKS
0: https://oeis.org/A012345/b012345.txt

FORMULAS
...
OFFSETS
(0, 1)

URL
https://oeis.org/A012345

AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)

Python

>>> from sage.all import *
>>> s = oeis(Integer(12345))                       # optional -- internet
>>> s.show()                              # optional -- internet
ID
A012345
<BLANKLINE>
NAME
Coefficients in the expansion sinh(arcsin(x)*arcsin(x)) = 2*x^2/2!+8*x^4/4!+248*x^6/6!+11328*x^8/8!+...
<BLANKLINE>
FIRST TERMS
(2, 8, 248, 11328, 849312, 94857600, 14819214720, 3091936512000, 831657655349760, 280473756197529600, 115967597965430077440, 57712257892456911912960, 34039765801079493369569280)
<BLANKLINE>
LINKS
0: https://oeis.org/A012345/b012345.txt
<BLANKLINE>
FORMULAS
...
OFFSETS
(0, 1)
<BLANKLINE>
URL
https://oeis.org/A012345
<BLANKLINE>
AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)
<BLANKLINE>

test_compile_sage_code()[source]¶

Try to compile the extracted sage code, if there is any.

If there are several sage code fields, they are all considered.

Dead sequences are considered to compile correctly by default.

This returns True if the code compiles, and raises an error otherwise.

EXAMPLES:

One correct sequence:

Sage

sage: s = oeis.find_by_id('A027642')        # optional -- internet
sage: s.test_compile_sage_code()            # optional -- internet
True

Python

>>> from sage.all import *
>>> s = oeis.find_by_id('A027642')        # optional -- internet
>>> s.test_compile_sage_code()            # optional -- internet
True

One dead sequence:

Sage

sage: s = oeis.find_by_id('A000154')        # optional -- internet
sage: s.test_compile_sage_code()            # optional -- internet
True

Python

>>> from sage.all import *
>>> s = oeis.find_by_id('A000154')        # optional -- internet
>>> s.test_compile_sage_code()            # optional -- internet
True

url()[source]¶

Return the URL of the page associated to the sequence self.

OUTPUT: string

EXAMPLES:

Sage

sage: f = oeis(45); f                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

sage: f.url()                               # optional -- internet
'https://oeis.org/A000045'

Python

>>> from sage.all import *
>>> f = oeis(Integer(45)); f                       # optional -- internet
A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

>>> f.url()                               # optional -- internet
'https://oeis.org/A000045'

sage.databases.oeis.to_tuple(string)[source]¶

Convert a string to a tuple of integers.

EXAMPLES:

Sage

sage: from sage.databases.oeis import to_tuple
sage: to_tuple('12,55,273')
(12, 55, 273)

Python

>>> from sage.all import *
>>> from sage.databases.oeis import to_tuple
>>> to_tuple('12,55,273')
(12, 55, 273)