CMO 1998 Problem 5

Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0, a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$ . Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$ , gives a solution to the equation $${ \frac{a^2 + b^2}{ab + 1} = m^2}$$ if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$ .