By the condition of the problem statement, we have $a_{n}^{2}-a_{n-1} a_{n+1}=1=a_{n-1}^{2}-a_{n-2} a_{n}$. This is equivalent to $\frac{a_{n-2}+a_{n}}{a_{n-1}}=\frac{a_{n-1}+a_{n+1}}{a_{n}}$. Let $k=\frac{a_{0}+a_{2}}{a_{1}}$. Then we have $\frac{a_{n-1}+a_{n+1}}{a_{n}}=\frac{a_{n-2}+a_{n}}{a_{n-1}}=\frac{a_{n-3}+a_{n-1}}{a_{n-2}}=\cdots=\frac{a_{0}+a_{2}}{a_{1}}=k$. Therefore we have $a_{n+1}=k a_{n}-a_{n-1}$ for all $n \geq 1$. We know that $k$ is a positive rational number because $a_{0}, a_{1}$, and $a_{2}$ are all positive integers. We claim that $k$ must be an integer. Suppose that $k=\frac{p}{q}$ with $\operatorname{gcd}(p, q)=1$. Since $k a_{n}=a_{n-1}+a_{n+1}$ is always an integer for $n \geq 1$, we must have $q \mid a_{n}$ for all $n \geq 1$. This contradicts $a_{2}^{2}-a_{1} a_{3}=1$. Conversely, if $k$ is an integer, inductively all $a_{i}$ are integers. Now we compute $a_{2}=\frac{t^{2}-1}{15}$, so $k=\frac{t^{2}+224}{15 t}$ is an integer. Therefore $15 k-t=\frac{224}{t}$ is an integer. Combining with the condition that $a_{2}$ is an integer limits the possible values of $t$ to $1,4,14,16,56$, 224. The values $t<15$ all lead to $a_{n}=0$ for some $n$ whereas $t>15$ leads to a good sequence. The sum of the solutions is $16+56+224=296$.
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.