$\begin{align*}
F_{n+1} &= \frac{2^{-1}(1+\sqrt{5})^{n-1}(1+\sqrt{5})^2 - 2^{-1}(1-\sqrt{5})^{n-1}(1-\sqrt{5})^2}{2^n\sqrt{5}} \\
&= \frac{(1+\sqrt{5})^{n+1} - (1-\sqrt{5})^{n+1}}{2^{n+1}\sqrt{5}}
\end{align*}$