BALKAN-SL 2015 N2

Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$ , and $a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$ . Prove that $n^2$ divides $a_n$ for infinite $n$ .

(Romania)