Does this sequence even exist?

Number Theory Level 5

Find sum of all positive integers $n$ such that there exist a sequence of positive integers $a_{1},a_{2},...,a_{n}$ satisfying $a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1$ for every $k$ with $2≤k≤n-1.$