Binary Recurrence Sequences

This class implements several methods relating to general linear binary recurrence sequences, including a sieve to find perfect powers in integral linear binary recurrence sequences.

EXAMPLES:

Sage

sage: R = BinaryRecurrenceSequence(1,1)        #the Fibonacci sequence
sage: R(137)        #the 137th term of the Fibonacci sequence
19134702400093278081449423917
sage: R(137) == fibonacci(137)
True
sage: [R(i) % 4 for i in range(12)]
[0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1]
sage: R.period(4)        #the period of the fibonacci sequence modulo 4
6
sage: R.pthpowers(2, 10**10)        # long time (7 seconds) -- in fact these are all squares, c.f. [BMS06]
[0, 1, 2, 12]

sage: S = BinaryRecurrenceSequence(8,1)  #a Lucas sequence
sage: S.period(73)
148
sage: S(5) % 73 == S(5 +148) %73
True
sage: S.pthpowers(3, 10**10)    # long time (3 seconds) -- provably finds the indices of all 3rd powers less than 10^10
[0, 1, 2]

sage: T = BinaryRecurrenceSequence(2,0,1,2)
sage: [T(i) for i in range(10)]
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
sage: T.is_degenerate()
True
sage: T.is_geometric()
True
sage: T.pthpowers(7, 10**30)                                                        # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: the degenerate binary recurrence sequence is geometric or quasigeometric
and has many pth powers

Python

>>> from sage.all import *
>>> R = BinaryRecurrenceSequence(Integer(1),Integer(1))        #the Fibonacci sequence
>>> R(Integer(137))        #the 137th term of the Fibonacci sequence
19134702400093278081449423917
>>> R(Integer(137)) == fibonacci(Integer(137))
True
>>> [R(i) % Integer(4) for i in range(Integer(12))]
[0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1]
>>> R.period(Integer(4))        #the period of the fibonacci sequence modulo 4
6
>>> R.pthpowers(Integer(2), Integer(10)**Integer(10))        # long time (7 seconds) -- in fact these are all squares, c.f. [BMS06]
[0, 1, 2, 12]

>>> S = BinaryRecurrenceSequence(Integer(8),Integer(1))  #a Lucas sequence
>>> S.period(Integer(73))
148
>>> S(Integer(5)) % Integer(73) == S(Integer(5) +Integer(148)) %Integer(73)
True
>>> S.pthpowers(Integer(3), Integer(10)**Integer(10))    # long time (3 seconds) -- provably finds the indices of all 3rd powers less than 10^10
[0, 1, 2]

>>> T = BinaryRecurrenceSequence(Integer(2),Integer(0),Integer(1),Integer(2))
>>> [T(i) for i in range(Integer(10))]
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
>>> T.is_degenerate()
True
>>> T.is_geometric()
True
>>> T.pthpowers(Integer(7), Integer(10)**Integer(30))                                                        # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: the degenerate binary recurrence sequence is geometric or quasigeometric
and has many pth powers

AUTHORS:

• Isabel Vogt (2013): initial version

See [SV2013], [BMS2006], and [SS1983].

class sage.combinat.binary_recurrence_sequences.BinaryRecurrenceSequence(b, c, u0=0, u1=1)

Bases: SageObject

Create a linear binary recurrence sequence defined by initial conditions \(u_0\) and \(u_1\) and recurrence relation \(u_{n+2} = b*u_{n+1}+c*u_n\).

INPUT:

• b – integer; (partially determining the recurrence relation)

• c – integer; (partially determining the recurrence relation)

• u0 – integer; (the \(0\)-th term of the binary recurrence sequence)

• u1 – integer; (the \(1\)-st term of the binary recurrence sequence)

OUTPUT: an integral linear binary recurrence sequence defined by \(u_0\), \(u_1\), and \(u_{n+2} = b u_{n+1}+c u_n\)

See also

fibonacci(), lucas_number1(), lucas_number2()

EXAMPLES:

Sage

sage: R = BinaryRecurrenceSequence(3,3,2,1)
sage: R
Binary recurrence sequence defined by: u_n = 3 * u_{n-1} + 3 * u_{n-2};
With initial conditions: u_0 = 2, and u_1 = 1

Python

>>> from sage.all import *
>>> R = BinaryRecurrenceSequence(Integer(3),Integer(3),Integer(2),Integer(1))
>>> R
Binary recurrence sequence defined by: u_n = 3 * u_{n-1} + 3 * u_{n-2};
With initial conditions: u_0 = 2, and u_1 = 1

is_arithmetic()

Decide whether the sequence is degenerate and an arithmetic sequence.

The sequence is arithmetic if and only if \(u_1 - u_0 = u_2 - u_1 = u_3 - u_2\).

This corresponds to the matrix \(F = [[0,1],[c,b]]\) being nondiagonalizable and \(\alpha/\beta = 1\).

EXAMPLES:

Sage

sage: S = BinaryRecurrenceSequence(2,-1)
sage: [S(i) for i in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: S.is_arithmetic()
True

Python

>>> from sage.all import *
>>> S = BinaryRecurrenceSequence(Integer(2),-Integer(1))
>>> [S(i) for i in range(Integer(10))]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> S.is_arithmetic()
True

is_degenerate()

Decide whether the binary recurrence sequence is degenerate.

Let \(\alpha\) and \(\beta\) denote the roots of the characteristic polynomial \(p(x) = x^2-bx -c\). Let \(a = u_1-u_0\beta/(\beta - \alpha)\) and \(b = u_1-u_0\alpha/(\beta - \alpha)\). The sequence is, thus, given by \(u_n = a \alpha^n - b\beta^n\). Then we say that the sequence is nondegenerate if and only if \(a*b*\alpha*\beta \neq 0\) and \(\alpha/\beta\) is not a root of unity.

More concretely, there are 4 classes of degeneracy, that can all be formulated in terms of the matrix \(F = [[0,1], [c, b]]\).

• \(F\) is singular – this corresponds to c = 0, and thus \(\alpha*\beta = 0\). This sequence is geometric after term u0 and so we call it quasigeometric

• \(v = [[u_0], [u_1]]\) is an eigenvector of \(F\) – this corresponds to a geometric sequence with \(a*b = 0\)

• \(F\) is nondiagonalizable – this corresponds to \(\alpha = \beta\). This sequence will be the point-wise product of an arithmetic and geometric sequence.

• \(F^k\) is scalar, for some \(k>1\) – this corresponds to \(\alpha/\beta\) a \(k\) th root of unity. This sequence is a union of several geometric sequences, and so we again call it quasigeometric.

EXAMPLES:

Sage

sage: S = BinaryRecurrenceSequence(0,1)
sage: S.is_degenerate()
True
sage: S.is_geometric()
False
sage: S.is_quasigeometric()
True

sage: R = BinaryRecurrenceSequence(3,-2)
sage: R.is_degenerate()
False

sage: T = BinaryRecurrenceSequence(2,-1)
sage: T.is_degenerate()
True
sage: T.is_arithmetic()
True

Python

>>> from sage.all import *
>>> S = BinaryRecurrenceSequence(Integer(0),Integer(1))
>>> S.is_degenerate()
True
>>> S.is_geometric()
False
>>> S.is_quasigeometric()
True

>>> R = BinaryRecurrenceSequence(Integer(3),-Integer(2))
>>> R.is_degenerate()
False

>>> T = BinaryRecurrenceSequence(Integer(2),-Integer(1))
>>> T.is_degenerate()
True
>>> T.is_arithmetic()
True

is_geometric()

Decide whether the binary recurrence sequence is geometric - ie a geometric sequence.

This is a subcase of a degenerate binary recurrence sequence, for which \(ab=0\), i.e. \(u_{n}/u_{n-1}=r\) for some value of \(r\).

See is_degenerate() for a description of degeneracy and definitions of \(a\) and \(b\).

EXAMPLES:

Sage

sage: S = BinaryRecurrenceSequence(2,0,1,2)
sage: [S(i) for i in range(10)]
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
sage: S.is_geometric()
True

Python

>>> from sage.all import *
>>> S = BinaryRecurrenceSequence(Integer(2),Integer(0),Integer(1),Integer(2))
>>> [S(i) for i in range(Integer(10))]
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
>>> S.is_geometric()
True

is_quasigeometric()

Decide whether the binary recurrence sequence is degenerate and similar to a geometric sequence, i.e. the union of multiple geometric sequences, or geometric after term u0.

If \(\alpha/\beta\) is a \(k\) th root of unity, where \(k>1\), then necessarily \(k = 2, 3, 4, 6\). Then \(F = [[0,1],[c,b]\) is diagonalizable, and \(F^k = [[\alpha^k, 0], [0,\beta^k]]\) is a diagonal matrix. Thus for all values of \(j\) mod \(k\), the \(j\) mod \(k\) terms of \(u_n\) form a geometric series.

If \(\alpha\) or \(\beta\) is zero, this implies that \(c=0\). This is the case when \(F\) is singular. In this case, \(u_1, u_2, u_3, ...\) is geometric.

EXAMPLES:

Sage

sage: S = BinaryRecurrenceSequence(0,1)
sage: [S(i) for i in range(10)]
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
sage: S.is_quasigeometric()
True

sage: R = BinaryRecurrenceSequence(3,0)
sage: [R(i) for i in range(10)]
[0, 1, 3, 9, 27, 81, 243, 729, 2187, 6561]
sage: R.is_quasigeometric()
True

Python

>>> from sage.all import *
>>> S = BinaryRecurrenceSequence(Integer(0),Integer(1))
>>> [S(i) for i in range(Integer(10))]
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
>>> S.is_quasigeometric()
True

>>> R = BinaryRecurrenceSequence(Integer(3),Integer(0))
>>> [R(i) for i in range(Integer(10))]
[0, 1, 3, 9, 27, 81, 243, 729, 2187, 6561]
>>> R.is_quasigeometric()
True

period(m)

Return the period of the binary recurrence sequence modulo an integer m.

If \(n_1\) is congruent to \(n_2\) modulo period(m), then \(u_{n_1}\) is is congruent to \(u_{n_2}\) modulo m.

INPUT:

• m – integer; modulo which the period of the recurrence relation is calculated

OUTPUT: integer (the period of the sequence modulo m)

EXAMPLES:

If \(p = \pm 1 \mod 5\), then the period of the Fibonacci sequence mod \(p\) is \(p-1\) (c.f. Lemma 3.3 of [BMS2006]).

Sage

sage: R = BinaryRecurrenceSequence(1,1)
sage: R.period(31)
30

sage: [R(i) % 4 for i in range(12)]
[0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1]
sage: R.period(4)
6

Python

>>> from sage.all import *
>>> R = BinaryRecurrenceSequence(Integer(1),Integer(1))
>>> R.period(Integer(31))
30

>>> [R(i) % Integer(4) for i in range(Integer(12))]
[0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1]
>>> R.period(Integer(4))
6

This function works for degenerate sequences as well.

Sage

sage: S = BinaryRecurrenceSequence(2,0,1,2)
sage: S.is_degenerate()
True
sage: S.is_geometric()
True
sage: [S(i) % 17 for i in range(16)]
[1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9]
sage: S.period(17)
8

Python

>>> from sage.all import *
>>> S = BinaryRecurrenceSequence(Integer(2),Integer(0),Integer(1),Integer(2))
>>> S.is_degenerate()
True
>>> S.is_geometric()
True
>>> [S(i) % Integer(17) for i in range(Integer(16))]
[1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9]
>>> S.period(Integer(17))
8

Note

The answer is cached.

pthpowers(p, Bound)

Find the indices of proveably all \(p\)-th powers in the recurrence sequence bounded by Bound.

Let \(u_n\) be a binary recurrence sequence. A p th power in \(u_n\) is a solution to \(u_n = y^p\) for some integer \(y\). There are only finitely many p th powers in any recurrence sequence [SS1983].

INPUT:

• p – a rational prime integer (the fixed p in \(u_n = y^p\))

• Bound – a natural number (the maximum index \(n\) in \(u_n = y^p\) that is checked)

OUTPUT:

A list of the indices of all p th powers less bounded by Bound. If the sequence is degenerate and there are many p th powers, raises ValueError.

EXAMPLES:

Sage

sage: R = BinaryRecurrenceSequence(1,1)        #the Fibonacci sequence
sage: R.pthpowers(2, 10**10)        # long time (7 seconds) -- in fact these are all squares, c.f. [BMS2006]_
[0, 1, 2, 12]

sage: S = BinaryRecurrenceSequence(8,1) #a Lucas sequence
sage: S.pthpowers(3,10**10)    # long time (3 seconds) -- provably finds the indices of all 3rd powers less than 10^10
[0, 1, 2]

sage: Q = BinaryRecurrenceSequence(3,3,2,1)
sage: Q.pthpowers(11,10**10)          # long time (7.5 seconds)
[1]

Python

>>> from sage.all import *
>>> R = BinaryRecurrenceSequence(Integer(1),Integer(1))        #the Fibonacci sequence
>>> R.pthpowers(Integer(2), Integer(10)**Integer(10))        # long time (7 seconds) -- in fact these are all squares, c.f. [BMS2006]_
[0, 1, 2, 12]

>>> S = BinaryRecurrenceSequence(Integer(8),Integer(1)) #a Lucas sequence
>>> S.pthpowers(Integer(3),Integer(10)**Integer(10))    # long time (3 seconds) -- provably finds the indices of all 3rd powers less than 10^10
[0, 1, 2]

>>> Q = BinaryRecurrenceSequence(Integer(3),Integer(3),Integer(2),Integer(1))
>>> Q.pthpowers(Integer(11),Integer(10)**Integer(10))          # long time (7.5 seconds)
[1]

If the sequence is degenerate, and there are no p th powers, returns \([]\). Otherwise, if there are many p th powers, raises ValueError.

Sage

sage: T = BinaryRecurrenceSequence(2,0,1,2)
sage: T.is_degenerate()
True
sage: T.is_geometric()
True
sage: T.pthpowers(7, 10**30)                                                # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: the degenerate binary recurrence sequence is geometric or
quasigeometric and has many pth powers

sage: L = BinaryRecurrenceSequence(4,0,2,2)
sage: [L(i).factor() for i in range(10)]
[2, 2, 2^3, 2^5, 2^7, 2^9, 2^11, 2^13, 2^15, 2^17]
sage: L.is_quasigeometric()
True
sage: L.pthpowers(2, 10**30)                                                # needs sage.symbolic
[]

Python

>>> from sage.all import *
>>> T = BinaryRecurrenceSequence(Integer(2),Integer(0),Integer(1),Integer(2))
>>> T.is_degenerate()
True
>>> T.is_geometric()
True
>>> T.pthpowers(Integer(7), Integer(10)**Integer(30))                                                # needs sage.symbolic
Traceback (most recent call last):
...
ValueError: the degenerate binary recurrence sequence is geometric or
quasigeometric and has many pth powers

>>> L = BinaryRecurrenceSequence(Integer(4),Integer(0),Integer(2),Integer(2))
>>> [L(i).factor() for i in range(Integer(10))]
[2, 2, 2^3, 2^5, 2^7, 2^9, 2^11, 2^13, 2^15, 2^17]
>>> L.is_quasigeometric()
True
>>> L.pthpowers(Integer(2), Integer(10)**Integer(30))                                                # needs sage.symbolic
[]

Note

This function is primarily optimized in the range where Bound is much larger than p.