Definition:General Fibonacci Sequence
Definition
Let $r, s, t, u$ be numbers, usually integers but not necessarily so limited.
Let $\sequence {a_n}$ be the sequence defined as:
- $a_n = \begin{cases}
r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$
Then $\sequence {a_n}$ is a general Fibonacci sequence.
Also defined as
Some sources define this with just $r$ and $s$ being the controllable parameters.
Such a sequence corresponds to this one where $t = u = 1$.
Also known as
A general Fibonacci sequence is usually referred to as a generalized Fibonacci sequence.
However, the spelling of generalized differs depending on what version of English is being used, so in order to ensure that $\mathsf{Pr} \infty \mathsf{fWiki}$ is as international as possible, the word general is preferred.
Also see
- Definition:Fibonacci Number: the general Fibonacci sequence where $r = 0, s = 1, t = 1, u = 1$
- Definition:Lucas Number: the general Fibonacci sequence where $r = 2, s = 1, t = 1, u = 1$
Source of Name
This entry was named for Leonardo Fibonacci.
Historical Note
The general Fibonacci sequences were first investigated by François Édouard Anatole Lucas, who used what are now known Fibonacci numbers and Lucas numbers to investigate the primality of Mersenne numbers.
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $12$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $11$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1,786,772,701,928,802,632,268,715,130,455,793$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $11$