Definition:Fibonacci-Like Sequence

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

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

& \\ \displaystyle \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:
 * $F_A \left({k}\right) = 2 F_A \left({k - 1}\right) - F_A \left({k - n}\right)$

Also see

 * Definition:Fibonacci Number, where $A = \left({0, 1}\right)$


 * Definition:Lucas Number, where $A = \left({2, 1}\right)$


 * Definition:General Fibonacci Sequence, where $A = \left({r, s}\right)$ for some numbers $r, s$


 * Definition:Tribonacci Sequence: $A = \left({0, 0, 1}\right)$


 * Definition:General Tribonacci Sequence: $A = \left({a, b, c}\right)$ for some numbers $a, b, c$


 * Definition:Keith Number