Fibonacci number

In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation:

$F_{n}:={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\F_{n-1}+F_{n-2}&{\mbox{if }}n>1.\\\end{cases}}$

The sequence of fibonacci numbers start: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

Fibonacci numbers and the rabbits

The sequence of fibonacci numbers was first used, to repesent the growth of a colony of rabbits, starting with one pair of rabbits.

Properties

• The quotient of two consecutive fibonacci numbers converges to the golden ratio: $\lim _{n\to \infty }{\frac {F(n+1)}{F(n)}}=\varphi$
• If $\scriptstyle m>2\$ divides $\scriptstyle n\$ then $\scriptstyle F_{m}\$ divides $\scriptstyle F_{n}\$
• If $\scriptstyle p\geq 5$ is a prime number, then is $\scriptstyle F_{p}\$ also a prime number.
• $\operatorname {gcd} (f_{m},f_{n})=f_{\operatorname {gcd} (m,n)}$
• $F_{0}^{2}+F_{1}^{2}+F_{2}^{2}+...+F_{n}^{2}=\sum _{i=0}^{n}F_{i}^{2}=F_{n}\cdot F_{n+1}$