Fibonacci Number by One Minus Golden Mean plus Fibonacci Number of Index One Less

Theorem
Let $n \in \Z$.

Then:


 * $\hat \phi^n = F_n \hat \phi + F_{n - 1}$

where:
 * $F_n$ denotes the $n$th Fibonacci number
 * $\hat \phi$ denotes the $1$ minus the golden mean:
 * $\hat \phi := 1 - \phi$

Also see

 * Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less