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)

[TAOCP1, p. 83], [Concrete, p. 299]:

$F_n = \frac{1}{\sqrt{5}} (\phi^n - \hat{\phi}^n), \quad n \in \mathbb{Z}, \quad \phi = \frac{1+\sqrt{5}}{2}, \hat{\phi} = \frac{1-\sqrt{5}}{2}$

(2)

[TAOCP1, p. 85]:

Fn + kFm − k − FnFm = ( − 1)nFm − n − kFk

(2)