KMO 2008 Problem 8

For fixed positive integers $s, t$ , define $a_n$ as the following. $a_1 = s, a_2 = t$ , and $\forall n \ge 1$ , $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$ . Prove that the solution set of $a_n \not= n$ , $n \in \mathbb{N}$ is finite.