Fibonacci Number of Odd Index by Golden Mean Modulo 1

Theorem
Let $n \in \Z$ be an integer.

Then:
 * $F_{2 n + 1} \phi \bmod 1 = \phi^{-2 n - 1}$

where:
 * $F_n$ denotes the $n$th Fibonacci number
 * $\phi$ is the golden mean: $\phi = \dfrac {1 + \sqrt 5} 2$

Proof
From definition of$\bmod 1$, the statement above is equivalent to the statement:
 * $F_{2 n + 1} \phi - \phi^{-2 n - 1}$ is an integer

We have:

Hence:

which is an integer.