Collection of Mathematical Formulas/Fibonacci Numbers

Definition of the Fibonacci sequence:

${\begin{cases}F_{0}=0\\F_{1}=1\\F_{n+2}=F_{n}+F_{n+1},&n\in \left[2,\infty \right)\end{cases}}$

(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]:

$F_{n+k}F_{m-k}-F_{n}F_{m}=(-1)^{n}F_{m-n-k}F_{k}$

(2)