BALKAN 2002 Problem 2

Let the sequence $\{a_n\}_{n\geq 1}$ be defined by $a_1 = 20$ , $a_2 = 30$ and $a_{n + 2} = 3a_{n + 1} - a_n$ for all $n\geq 1$ . Find all positive integers $n$ such that $1 + 5a_n a_{n + 1}$ is a perfect square.