Why is $ \frac{5}{64}((161+72\sqrt{5})^{-n}+(161+72\sqrt{5})^{n}-2)$ always a perfect square? I'm working on a puzzle, and the solution requires me somehow establishing that
$$ f(n):=\frac{5}{64}\Big(\big(161+72\sqrt{5}\big)^{-n}+\big(161+72\sqrt{5}\big)^{n}-2\Big)$$
is a perfect square for $n\in \mathbb{Z}_{\geq 0}$.
I've done a lot of simplification to get to this point, and am stuck here. I can provide the context of the puzzle if necessary, but it's pretty far removed from what I have here. The goal is basically to show that a formula generates solutions to a given equation.
Any tips on how to proceed?
Here's the first few values:
$$\begin{array}{|c|c|}
\hline
n&\text{value}\\ \hline
0&0\\ \hline
1& 5^2 \\ \hline
 2&90^2 \\ \hline
 3& 1615^2\\ \hline
 4& 28980^2\\ \hline
\end{array}$$

It's $$\frac{5}{64}\left((9+4\sqrt5)^{2n}+(9+4\sqrt5)^{-2n}-2\right)=\frac{5}{64}\left((9+4\sqrt5)^n-(9-4\sqrt5)^n\right)^2.$$
Can you end it now?
