Mathematics

This is a functional equation problem where you need to find a pattern by using the recurrence relation strategically. Start by computing small values like $a_2, a_3, a_4$ using the given rule with clever choices of $m$ and $n$ (like setting one equal to 1). Once you spot the pattern, you can either guess a closed form for $a_n$ and verify it satisfies the equation, or continue building up to $a_{12}$ directly. The key insight is that the $mn$ term suggests $a_n$ likely has a quadratic component.

This is a functional equation problem where you need to find a pattern by using the recurrence relation strategically. Start by computing small values like $a_2, a_3, a_4$ using the given rule with clever choices of $m$ and $n$ (like setting one equal to 1). Once you spot the pattern, you can either guess a closed form for $a_n$ and verify it satisfies the equation, or continue building up to $a_{12}$ directly. The key insight is that the $mn$ term suggests $a_n$ likely has a quadratic component.