Collection of Mathematical Formulas/Fibonacci Numbers

Definition of the Fibonacci sequence:

$\{\begin{array}{c}{F}_0=0\\ F_1=1\\ F_{n+2}=F_n+F_{n+1},& n\in [2,\mathrm{\infty })\end{array}$

(1)

[TAOCP1, p. 83], [Concrete, p. 299]:

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

(2)

[TAOCP1, p. 85]:

$F_{n+k}{F}_{m-k}-F_n{F}_m=(-1{)}^n{F}_{m-n-k}{F}_k$

(2)