MathNet 2010 — Part 07 Question 72

Let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers such that for integers $n \gt 2$, $a_{n} = 3 a_{n-1} - 2 a_{n-2}$. How many such sequences $\{a_{n}\}$ are there such that $a_{2010} \leq 2^{2012}$?