# 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)

>>> 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