Details

FindLinearRecurrence gives the shortest kernel ker which reproduces list from LinearRecurrence[ker,init,…]. The initial list init comes from the first elements in list.
FindLinearRecurrence[list] will find the coefficients {a₁,…,a_d} such that LinearRecurrence[{a₁,…,a_d},list[[1;;d]],Length[list]] generates list.

Examples

Find linear recurrences for simple sequences:

Wolfram Language code: FindLinearRecurrence[Table[2 ^ k, {k, 10}]]

Wolfram Language code: FindLinearRecurrence[Table[Fibonacci[k], {k, 10}]]

Wolfram Language code: FindLinearRecurrence[Table[Fibonacci[k] Fibonacci[k - 1] ^ 2, {k, 10}]]

Generate data from a linear recurrence:

Wolfram Language code: data = LinearRecurrence[{4, 3, 2, 1}, {1, 2, 3, 4}, 10]

Find the underlying recurrence equation:

Wolfram Language code: FindLinearRecurrence[data]

Use symbolic data:

Wolfram Language code: data = Simplify@LinearRecurrence[{4, 3, 2, 1}, {a, b, c, d}, 10]

Wolfram Language code: FindLinearRecurrence[data]

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