BALKAN 1986 Problem 3

Let $a,b,c$ be real numbers such that $ab\not= 0$ and $c>0$ . Let $(a_{n})_{n\geq 1}$ be the sequence of real numbers defined by: $a_{1}=a, a_{2}=b$ and $$a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}$$ for all $n\geq 2$ . Show that all the terms of the sequence are integer numbers if and only if the numbers $a,b$ and $\frac{a^{2}+b^{2}+c}{ab}$ are integers.