# 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): $11$