ISL 2009 N4

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