Euler-Binet Formula

Theorem
The Fibonacci numbers have a closed-form solution:
 * $F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n} {\phi - \paren {1 - \phi}}$

where $\phi$ is the golden mean.

Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written:
 * $F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$

Putting $\sqrt 5 = 2\phi - 1 = \phi - \paren {1 - \phi} = \phi - \hat \phi$ this can be written:
 * $F_n = \dfrac {\phi^n - \hat \phi^n} {\paren {\phi - \hat \phi}}$