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)

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)

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