$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$ where $\phi = \frac{1+\sqrt{5}}{2}, \psi = \frac{1-\sqrt{5}}{2}$

$F_n = \frac{\phi^n + \psi^n}{\sqrt{5}}$

$F_n = \phi^n - \psi^n$

$F_n = \frac{\phi^n - \psi^n}{2}$