Lucas Number 2n in terms of Square of Lucas Number n

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

Then:
 * $L_{2 n} = {L_n}^2 + 2 \left({-1}\right)^n$

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

Note that we have:

Then:

Hence the result.