FindSequenceFunction

A closed form $a(n)$ for a sequence given as a list $a(0), a(1), \dots$ , when one can be found.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFindSequenceFunction!

FindSequenceFunction(list, n)a closed form for the sequence given by list, in n

Typed signatures

Type	Package	Applies
`(list<any>, symbol) -> any`	`combinatorics`	always

Domain: Combinatorics

Details

Tries, in order, a polynomial fit (exact finite differences: the least degree whose difference row vanishes), a small registry of named sequences (Fibonacci, Catalan, factorial, derangements, Bell numbers) matched by value, and a linear-recurrence fit (Berlekamp--Massey over ℚ) resolved to a closed form for orders 1 (geometric) and 2 (exponential-polynomial, via the recurrence's characteristic roots).
Every closed form found is exact -- re-evaluating it reproduces the input list precisely, not just approximately.
Falls back to staying symbolic when nothing fits.

Examples

Basic #

$2^n$

In

\mathrm{FindSequenceFunction}(\bigl\lbrack1, 2, 4, 8, 16\bigr\rbrack, n)

Out

2^{n}

$(n+1)^2$ , found by finite differences

In

\mathrm{FindSequenceFunction}(\bigl\lbrack1, 4, 9, 16, 25\bigr\rbrack, n)

Out

n^2+2n+1

recognised by value against the compute engine's own CatalanNumber

In

\mathrm{FindSequenceFunction}(\bigl\lbrack1, 1, 2, 5, 14, 42, 132\bigr\rbrack, n)

Out

\mathrm{CatalanNumber}(n)

$n!$ -- not C-finite, so only reached via the named-sequence registry, not Berlekamp--Massey

In

\mathrm{FindSequenceFunction}(\bigl\lbrack1, 1, 2, 6, 24, 120\bigr\rbrack, n)

Out

n!

In

\mathrm{FindSequenceFunction}(\bigl\lbrack1, 1, 2, 3, 5, 8, 13\bigr\rbrack, n)

Out

\frac{1}{10}(5-\sqrt{5})(\frac{1-5^{1/2}}{2})^{n}+\frac{1}{10}\sqrt{5}(1+\sqrt{5})(\frac{1+\sqrt{5}}{2})^{n}

Scope #

In

\mathrm{FindSequenceFunction}(\bigl\lbrack-1, 1, -1, 1, -1, 1\bigr\rbrack, n)

Out

-(-1)^{n}

In

\mathrm{FindSequenceFunction}(\bigl\lbrack2, 5, 10, 17, 26, 37, 50\bigr\rbrack, n)

Out

n^2+2n+2

Properties #

In

\mathrm{FindSequenceFunction}(\bigl\lbrack3, 7, 13, 21, 31, 43, 57\bigr\rbrack, n)

Out

n^2+3n+3

About

Examples	8
Checked against	Wolfram Language 8 differ
Crosswalk	Wolfram Language

See also: GeneratingFunction, ExponentialGeneratingFunction, DiscreteRatio