\displaystyle \sum_{n \geq 0}a_{n+2}x^n = \sum_{n \geq 0}a_{n+1}x^n + 2 \sum_{n \geq 0}a_n x^n + \sum_{n \geq 0}x^n \\
\Longleftrightarrow \frac{A(x) - a_1x - a_0}{x^2} = \frac{A(x) - a_0}{x} + 2 A(x) + \frac{1}{1 - x} \\
\Longleftrightarrow A(x) \frac{1 - x - 2x^2}{x^2} = \frac{1}{x} + \frac{1}{1-x}