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 \(2≤k≤n-1.\)