# Sequences¶

A sequence is a finite or infinite lazily evaluated list.

sympy.series.sequences.sequence(seq, limits=None)[source]

Returns appropriate sequence object.

Explanation

If seq is a SymPy sequence, returns SeqPer object otherwise returns SeqFormula object.

Examples

>>> from sympy import sequence
>>> from sympy.abc import n
>>> sequence(n**2, (n, 0, 5))
SeqFormula(n**2, (n, 0, 5))
>>> sequence((1, 2, 3), (n, 0, 5))
SeqPer((1, 2, 3), (n, 0, 5))


## Sequences Base¶

class sympy.series.sequences.SeqBase(*args)[source]

Base class for sequences

coeff(pt)[source]

Returns the coefficient at point pt

coeff_mul(other)[source]

Should be used when other is not a sequence. Should be defined to define custom behaviour.

Examples

>>> from sympy import SeqFormula
>>> from sympy.abc import n
>>> SeqFormula(n**2).coeff_mul(2)
SeqFormula(2*n**2, (n, 0, oo))


Notes

‘*’ defines multiplication of sequences with sequences only.

find_linear_recurrence(n, d=None, gfvar=None)[source]

Finds the shortest linear recurrence that satisfies the first n terms of sequence of order $$\leq$$ n/2 if possible. If d is specified, find shortest linear recurrence of order $$\leq$$ min(d, n/2) if possible. Returns list of coefficients [b(1), b(2), ...] corresponding to the recurrence relation x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ... Returns [] if no recurrence is found. If gfvar is specified, also returns ordinary generating function as a function of gfvar.

Examples

>>> from sympy import sequence, sqrt, oo, lucas
>>> from sympy.abc import n, x, y
>>> sequence(n**2).find_linear_recurrence(10, 2)
[]
>>> sequence(n**2).find_linear_recurrence(10)
[3, -3, 1]
>>> sequence(2**n).find_linear_recurrence(10)
[2]
>>> sequence(23*n**4+91*n**2).find_linear_recurrence(10)
[5, -10, 10, -5, 1]
>>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10)
[1, 1]
>>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30)
[1/2, 1/2]
>>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x)
([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1)))
>>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x)
([1, 1], (x - 2)/(x**2 + x - 1))

property free_symbols

This method returns the symbols in the object, excluding those that take on a specific value (i.e. the dummy symbols).

Examples

>>> from sympy import SeqFormula
>>> from sympy.abc import n, m
>>> SeqFormula(m*n**2, (n, 0, 5)).free_symbols
{m}

property gen

Returns the generator for the sequence

property interval

The interval on which the sequence is defined

property length

Length of the sequence

property start

The starting point of the sequence. This point is included

property stop

The ending point of the sequence. This point is included

property variables

Returns a tuple of variables that are bounded

## Elementary Sequences¶

class sympy.series.sequences.SeqFormula(formula, limits=None)[source]

Represents sequence based on a formula.

Elements are generated using a formula.

Examples

>>> from sympy import SeqFormula, oo, Symbol
>>> n = Symbol('n')
>>> s = SeqFormula(n**2, (n, 0, 5))
>>> s.formula
n**2


For value at a particular point

>>> s.coeff(3)
9


supports slicing

>>> s[:]
[0, 1, 4, 9, 16, 25]


iterable

>>> list(s)
[0, 1, 4, 9, 16, 25]


sequence starts from negative infinity

>>> SeqFormula(n**2, (-oo, 0))[0:6]
[0, 1, 4, 9, 16, 25]

coeff_mul(coeff)[source]

See docstring of SeqBase.coeff_mul

class sympy.series.sequences.SeqPer(periodical, limits=None)[source]

Represents a periodic sequence.

The elements are repeated after a given period.

Examples

>>> from sympy import SeqPer, oo
>>> from sympy.abc import k

>>> s = SeqPer((1, 2, 3), (0, 5))
>>> s.periodical
(1, 2, 3)
>>> s.period
3


For value at a particular point

>>> s.coeff(3)
1


supports slicing

>>> s[:]
[1, 2, 3, 1, 2, 3]


iterable

>>> list(s)
[1, 2, 3, 1, 2, 3]


sequence starts from negative infinity

>>> SeqPer((1, 2, 3), (-oo, 0))[0:6]
[1, 2, 3, 1, 2, 3]


Periodic formulas

>>> SeqPer((k, k**2, k**3), (k, 0, oo))[0:6]
[0, 1, 8, 3, 16, 125]

coeff_mul(coeff)[source]

See docstring of SeqBase.coeff_mul

## Singleton Sequences¶

class sympy.series.sequences.EmptySequence[source]

Represents an empty sequence.

The empty sequence is also available as a singleton as S.EmptySequence.

Examples

>>> from sympy import EmptySequence, SeqPer
>>> from sympy.abc import x
>>> EmptySequence
EmptySequence
>>> SeqPer((1, 2), (x, 0, 10)) + EmptySequence
SeqPer((1, 2), (x, 0, 10))
>>> SeqPer((1, 2)) * EmptySequence
EmptySequence
>>> EmptySequence.coeff_mul(-1)
EmptySequence

coeff_mul(coeff)[source]

See docstring of SeqBase.coeff_mul

## Compound Sequences¶

Rules:
• The interval on which sequence is defined is the intersection of respective intervals of sequences.

• Anything + EmptySequence remains unchanged.

• Other rules are defined in _add methods of sequence classes.

Examples

>>> from sympy import EmptySequence, oo, SeqAdd, SeqPer, SeqFormula
>>> from sympy.abc import n
>>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), EmptySequence)
SeqPer((1, 2), (n, 0, oo))
>>> SeqAdd(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10)))
EmptySequence
>>> SeqAdd(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2, (n, 0, oo)))
SeqAdd(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo)))
SeqFormula(n**3 + n**2, (n, 0, oo))

static reduce(args)[source]

Simplify SeqAdd using known rules.

Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent.

Notes

adapted from Union.reduce

class sympy.series.sequences.SeqMul(*args, **kwargs)[source]

Represents term-wise multiplication of sequences.

Explanation

Handles multiplication of sequences only. For multiplication with other objects see SeqBase.coeff_mul().

Rules:

Examples

>>> from sympy import EmptySequence, oo, SeqMul, SeqPer, SeqFormula
>>> from sympy.abc import n
>>> SeqMul(SeqPer((1, 2), (n, 0, oo)), EmptySequence)
EmptySequence
>>> SeqMul(SeqPer((1, 2), (n, 0, 5)), SeqPer((1, 2), (n, 6, 10)))
EmptySequence
>>> SeqMul(SeqPer((1, 2), (n, 0, oo)), SeqFormula(n**2))
SeqMul(SeqFormula(n**2, (n, 0, oo)), SeqPer((1, 2), (n, 0, oo)))
>>> SeqMul(SeqFormula(n**3), SeqFormula(n**2))
SeqFormula(n**5, (n, 0, oo))

static reduce(args)[source]

Simplify a SeqMul using known rules.

Explanation

Iterates through all pairs and ask the constituent sequences if they can simplify themselves with any other constituent.

Notes

adapted from Union.reduce

## Recursive Sequences¶

class sympy.series.sequences.RecursiveSeq(recurrence, yn, n, initial=None, start=0)[source]

A finite degree recursive sequence.

Parameters:

recurrence : SymPy expression defining recurrence

This is not an equality, only the expression that the nth term is equal to. For example, if a(n) = f(a(n - 1), ..., a(n - d)), then the expression should be f(a(n - 1), ..., a(n - d)).

yn : applied undefined function

Represents the nth term of the sequence as e.g. y(n) where y is an undefined function and $$n$$ is the sequence index.

n : symbolic argument

The name of the variable that the recurrence is in, e.g., n if the recurrence function is y(n).

initial : iterable with length equal to the degree of the recurrence

The initial values of the recurrence.

start : start value of sequence (inclusive)

Explanation

That is, a sequence a(n) that depends on a fixed, finite number of its previous values. The general form is

a(n) = f(a(n - 1), a(n - 2), …, a(n - d))

for some fixed, positive integer d, where f is some function defined by a SymPy expression.

Examples

>>> from sympy import Function, symbols
>>> from sympy.series.sequences import RecursiveSeq
>>> y = Function("y")
>>> n = symbols("n")
>>> fib = RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, [0, 1])

>>> fib.coeff(3) # Value at a particular point
2

>>> fib[:6] # supports slicing
[0, 1, 1, 2, 3, 5]

>>> fib.recurrence # inspect recurrence
Eq(y(n), y(n - 2) + y(n - 1))

>>> fib.degree # automatically determine degree
2

>>> for x in zip(range(10), fib): # supports iteration
...     print(x)
(0, 0)
(1, 1)
(2, 1)
(3, 2)
(4, 3)
(5, 5)
(6, 8)
(7, 13)
(8, 21)
(9, 34)

property initial

The initial values of the sequence

property interval

Interval on which sequence is defined.

property n

Sequence index symbol

property recurrence

Equation defining recurrence.

property start

The starting point of the sequence. This point is included

property stop

The ending point of the sequence. (oo)

property y

Undefined function for the nth term of the sequence

property yn

Applied function representing the nth term