CZ-SK-MO 1986 Problem 5

A sequence of natural numbers $a_1,a_2,...$ satisfies $a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2$ for $n \in N$ . Prove that for every natural $n$ there exists a natural $m$ such that $a_na_{n+1} = a_m$ .