IR-MO 2002 Problem 3

$a_{n}$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3$ , and $$a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}$$ Prove that for each natural $n$ , $a_{n}$ is integer.