Question 29
Let $m$ be a positive integer. Define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ by $a_{0}=0$, $a_{1}=m$, and $a_{n+1}=m^{2} a_{n}-a_{n-1}$ for $n=1,2,3, \ldots$. Prove that an ordered pair $(a, b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation
$$
\frac{a^{2}+b^{2}}{a b+1}=m^{2}
$$
if and only if $(a, b)$ is of the form $\left(a_{n}, a_{n+1}\right)$ for some $n \geq 0$.
$$
\frac{a^{2}+b^{2}}{a b+1}=m^{2}
$$
if and only if $(a, b)$ is of the form $\left(a_{n}, a_{n+1}\right)$ for some $n \geq 0$.