Find all positive integers
such that there exists a sequence of positive integers
,
, ...,
satisfying
for every
with
.
Proposed by North Korea
%V0
Find all positive integers $n$ such that there exists 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 $2 \leqslant k \leqslant n-1$.
Proposed by North Korea
Školjka