Fibonacci Number of Index 3n as Sum of Cubes of Fibonacci Numbers

Theorem
Let $F_n$ denote the $n$th Fibonacci number.

Then:
 * $F_{3 n} = {F_{n + 1} }^3 + {F_n}^3 - {F_{n - 1} }^3$

Proof
From Fibonacci Number in terms of Smaller Fibonacci Numbers:


 * $\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

Setting $m = 2 n$: