HMMT 2020 U21

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

$$a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)$$

for all positive integers $n$. Find the sum of all good numbers.