HMMT November 2010 U23

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