A recurrence is defined as $a_n = a_{n-1} + 2a_{n-2}$ for $n \geq 2$, with $a_0 = 1$ and $a_1 = 1$. Let $A(x) = \sum_{n=0}^{\infty} a_n x^n$ be the ordinary generating function. Which of the following is the correct rational form of $A(x)$?