Eccoci.

Let $r$ be the remainder when $(a - 1)^n + (a + 1)^n$ is divided by $a^2$.

For example, if $a = 7$ and $n = 3$, then $r = 42$: $6^3 + 8^3 = 728 \equiv 42 \mod 49$. And as $n$ varies, so too will $r$, but for $a = 7$ it turns out that $r_{\mathrm{max} } = 42$.

For $3 \le a \le 1000$, find $\sum r_{\mathrm{max} }$.