Definition:Fibonacci-Like Sequence

Definition
Let $A = \tuple {a_0, a_1, \ldots, a_{n - 1} }$ be an ordered tuple of numbers.

The Fibonacci-like sequence formed from $A$ is defined as:
 * $\map {F_A} k = \begin {cases} \qquad \qquad a_k & : 0 \le k < n \\

& \\ \ds \sum_{k - n \mathop \le j \mathop < k} a_j & : k \ge n \end {cases}$

That is, apart from the first $n$ terms, every term is the sum of the previous $n$ terms.

The main term can also be expressed as:
 * $\map {F_A} k = 2 \map {F_A} {k - 1} - \map {F_A} {k - n}$

Also see

 * Definition:Fibonacci Number, where $A = \tuple {0, 1}$


 * Definition:Lucas Number, where $A = \tuple {2, 1}$


 * Definition:General Fibonacci Sequence, where $A = \tuple {r, s}$ for some numbers $r, s$


 * Definition:Tribonacci Sequence: $A = \tuple {0, 0, 1}$


 * Definition:General Tribonacci Sequence: $A = \tuple {a, b, c}$ for some numbers $a, b, c$


 * Definition:Keith Number