2009 IMO Shortlist N4

PRK 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 $2 \leq k \leq n-1$.

PRK 找到所有正整数 $n$，使得对于每个 $k$，存在满足 $$a_{k+1}=\frac{a_{k}^{2}+1}{a_{k-1}+1}-1$$ 且 $2 \leq k \leq n-1$ 的正整数序列 $a_{1}, a_{2}, \ldots, a_{n}$。