Fibonacci Number 3n in terms of Fibonacci Number n and Lucas Number 2n

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

Let $L_n$ denote the $n$th Lucas number.

Then:
 * $F_{3 n} = F_n \paren {L_{2 n} + \paren {-1}^n}$

Proof
Let:
 * $\phi = \dfrac {1 + \sqrt 5} 2$
 * $\hat \phi = \dfrac {1 - \sqrt 5} 2$

Then:

Then we note:

The result follows.