Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

The $n$th Fibonacci number $F_n$ is defined recursively, by

$$F_n = F_{n - 1} + F_{n - 2}$$

for $n > 1$, and $F_0 = 0,\; F_1 = 1$. There is a closed form expression, namely

$$F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}$$

where the golden ratio $\varphi$ is equal to $\frac{1 + \sqrt{5}}{2}$.

Combinatorial identities involving the Fibonacci numbers have been extensively studied, and the numbers arise frequently in nature and in popular culture.

Reference: Fibonacci number.

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