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:
  • FunctionSpaceAutomaticwhere to look for candidate simple functions
    MethodAutomaticmethod to use
    TimeConstraint10how many seconds to search a particular function space or perform a transformation
    ValidationLengthAutomaticsequence 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

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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:

FindSequenceFunction assumes that sequences start with index 1:

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:

Use FindGeneratingFunction:

Wolfram Research (2008), FindSequenceFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/FindSequenceFunction.html (updated 2015).

Text

Wolfram Research (2008), FindSequenceFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/FindSequenceFunction.html (updated 2015).

CMS

Wolfram Language. 2008. "FindSequenceFunction." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/FindSequenceFunction.html.

APA

Wolfram Language. (2008). FindSequenceFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindSequenceFunction.html

BibTeX

@misc{reference.wolfram_2024_findsequencefunction, author="Wolfram Research", title="{FindSequenceFunction}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FindSequenceFunction.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_findsequencefunction, organization={Wolfram Research}, title={FindSequenceFunction}, year={2015}, url={https://reference.wolfram.com/language/ref/FindSequenceFunction.html}, note=[Accessed: 21-December-2024 ]}