# FindSequenceFunction

FindSequenceFunction[{a1,a2,a3,}]

attempts to find a simple function that yields the sequence an when given successive integer arguments.

FindSequenceFunction[{{n1,a1},{n2,a2},}]

attempts to find a simple function that yields ai when given argument ni.

FindSequenceFunction[n1a1,n2a2,]

gives a function that yields ai when given argument ni.

FindSequenceFunction[{n1a1,n2a2,}]

gives a function that yields ai when given argument ni.

FindSequenceFunction[list,n]

gives the function applied to n.

# Details and Options • The sequence elements an can be either exact numbers or symbolic expressions.
• FindSequenceFunction finds results in terms of a wide range of integer functions, as well as implicit solutions to difference equations represented by DifferenceRoot.
• If FindSequenceFunction cannot find a simple function that yields the specified sequence, it returns unevaluated.
• The following options can be used:
•  FunctionSpace Automatic where to look for candidate simple functions Method Automatic method to use TimeConstraint 10 how many seconds to search a particular function space or perform a transformation ValidationLength Automatic sequence length used to validate a candidate function found
• FindSequenceFunction[list] by default uses earlier elements in list to find candidate simple functions, then validates the functions by looking at later elements.
• FindSequenceFunction[list] only returns functions that correctly reproduce all elements of list.

# Examples

open allclose all

## Basic Examples(2)

Find a sequence that yields the sequence 1,1,2,3,5,8,13,:

Find a function that yields the given sequence as a subsequence:

Check the even subsequence:

## Scope(5)

Periodic sequences:

Polynomial functions:

Rational functions:

Hypergeometric terms:

Recurrence equations:

## Generalizations & Extensions(1)

FindSequenceFunction works on arbitrary exact numbers or symbolic expressions:

## Applications(6)

Find formulas for complex sequences:

Use additional values to validate the result:

Find a closed form for a sequence of definite integrals:

Find a closed form for the number of 0,1 sequences of length containing two adjacent 1s:

Generate a sequence from a power series expansion:

Find its formula:

Use SeriesCoefficient to find an alternative formula:

Compare the result:

Compute a finite number of Fourier coefficients:

Find the formula:

Use a FourierCoefficient directly:

Verify the consistency of formulas:

Construct the Cantor set by starting with a {0,1} interval and removing the middle third of each interval in each step:

Some steps:

Find the length of the region:

Find a formula for the sequence of lengths using FindSequenceFunction:

## Properties & Relations(2)

Sum, Product, and other general discrete functions may be used:

Find the generating function of a sequence: