Let $A(x) = \sum_{n=0}^{\infty} a_n x^n$ be the generating function for a sequence satisfying $a_n = 2a_{n-1} + a_{n-2} - 2a_{n-3}$ with initial conditions $a_0 = 1, a_1 = 1, a_2 = 2$. What is the expression for $A(x)$?