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

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $14$

• Earls, Jason, Lichtblau, Daniel and Weisstein, Eric W. "Keith Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KeithNumber.html