# Collection of Mathematical Formulas/Fibonacci Numbers

Definition of the Fibonacci sequence:

 $\begin{cases} F_0 = 0 \\ F_1 = 1 \\ F_{n+2} = F_n+F_{n+1}, &n = 0, 1, 2, 3, 5, 8 ... \end{cases}$ (1)
 $F_n = \frac{1}{\sqrt{5}} (\phi^n - \hat{\phi}^n), \quad n \in \mathbb{Z}, \quad \phi = \frac{1+\sqrt{5}}{2}, \hat{\phi} = \frac{1-\sqrt{5}}{2}$ (2)
 $F_{n+k} F_{m-k} - F_n F_m = (-1)^n F_{m-n-k} F_k$ (2)