Question 3
Let $k \geqslant 2$ be an integer. Determine all sequences of positive integers $a_{1}, a_{2}, \ldots$ for which there exists a monic polynomial $P$ of degree $k$ with non-negative integer coefficients such that
$$
P\left(a_{n}\right)=a_{n+1} a_{n+2} \cdots a_{n+k}
$$
for every integer $n \geqslant 1$.
$$
P\left(a_{n}\right)=a_{n+1} a_{n+2} \cdots a_{n+k}
$$
for every integer $n \geqslant 1$.