Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer. How do we prove that $\dfrac{\phi^{400}+1}{\phi^{200}}$ is an integer, where $\phi$ is the golden ratio?
This appeared in an answer to a question I asked previously, but I do not see how to prove this..

More generally,
$\dfrac{\phi^{2n}+1}{\phi^{n}}$ is an integer for $n$ even.
Indeed, let $n=2m$ and $\alpha=\phi^2$. Then
$$
\dfrac{\phi^{2n}+1}{\phi^{n}}
= \phi^{n}+\dfrac{1}{\phi^{n}}
= \alpha^{m}+\dfrac{1}{\alpha^{m}}
=: y_m
$$
Since $\alpha$ and $\dfrac{1}{\alpha}$ are the roots of $x^2=3x-1$, we have
$ y_{k+2} = 3y_{k+1}-y_{k}$ for all $k \in \mathbb N$.
Since $y_0=2$ and $y_1=3$ are integers, so is $y_k$ for every $k \in \mathbb N$.
In fact, $y_k=L_{2k}$, the $2k$-th Lucas number.
Yet more generally,
$\dfrac{\phi^{2n}+(-1)^n}{\phi^{n}}$ is an integer for all $n \in \mathbb N$. In fact, it is the  $n$-th Lucas number.
