Question 64
We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \ldots$ of positive integers satisfying $a_{0}=15$, $a_{1}=t$, and
$$
a_{n-1} a_{n+1} = (a_{n} - 1)(a_{n} + 1)
$$
for all positive integers $n$. Find the sum of all good numbers.
$$
a_{n-1} a_{n+1} = (a_{n} - 1)(a_{n} + 1)
$$
for all positive integers $n$. Find the sum of all good numbers.