Can this be simplified any further?:
$\phi^{2n}$ + $\psi^{2n}$$-2(-1)^n$ $\pm4$
Where $\phi = (1+\sqrt5)/2 $
Where $\psi = (1-\sqrt5)/2 $
When n=even number, use +4
Whenn n=odd number, use -4
If anyone could show me how to simplify this further it would be greatly appreciated.
You are dealing with Lucas numbers, defined by $L_0=2$, $L_1=1$, $L_n=L_{n-1}+L_{n-2}$. We have $L_n=\varphi^n+\psi^n$.
Observe that $L_n^2=\varphi^{2n}+2\varphi^n\psi^n+\psi^{2n}=L_{2n}+2(-1)^n$.