Solution:

Answer: $36 \cdot 2^{2009} + 36$

Consider the characteristic polynomial for the recurrence $a_{n+2} - 3 a_{n+1} + 2 a_{n} = 0$, which is $x^{2} - 3x + 2$. The roots are at $2$ and $1$, so we know that numbers $a_{i}$ must be of the form $a 2^{i-1} + b$ for integers $a$ and $b$. Therefore $a_{2010}$ must equal $a 2^{2009} + b$, where $a$ and $b$ are both integers. If the expression is always positive, it is sufficient to say $a_{1}$ is positive and $a$ is nonnegative, or $a + b > 0$, and $a \geq 0$.

For a given value of $a$, $1 - a \leq b \leq 2^{2012} - a 2^{2009}$, so there are $2^{2012} - a 2^{2009} + a$ possible values of $b$ for each $a$ (where the quantity is positive). $a$ can take any value between $0$ and $2^{3}$, we sum over all such $a$ in this range, to attain $9 \cdot 2^{2012} - (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) 2^{2009} + (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)$, or $36\left(2^{2009}\right) + 36$, which is our answer.