Write the generating function of a sequence as the quotient of two polynomials

Question

I want to write the generating function of the sequence

$$a_0=2, \;a_1=3, \; a_{n}=2a_{n-1}-2a_{n-2} \; \text{for }n\geq 2$$

as the quotient of two polynomials. But how can I do that? What is the best way?

Answer 1

Just use the definition of the generating function: \begin{align*} G(x)&=\sum_{n=0}^{\infty}a_nx_n=a_0+a_1x+\sum_{n=2}^{\infty}a_nx^n \\ &=a_0+a_1x+\sum_{n=2}^{\infty}(2a_{n-1}-2a_{n-2})x^n \\ &=a_0+a_1x+2\sum_{n=1}^{\infty}a_nx^{n+1}-2\sum_{n=0}^{\infty}a_nx^{n+2} \\ &=a_0+a_1x+2x\left(-a_0+\sum_{n=0}^{\infty}a_nx^n\right)-2x^2\sum_{n=0}^{\infty}a_nx^n \\ &=a_0+(a_1-2a_0)x+2xG(x)-2x^2G(x) \end{align*} Solving for $G(x)$ we obtain $$ G(x)=\frac{a_0+(a_1-2a_0)x}{1-2x+2x^2}=\frac{2-x}{1-2x+2x^2}. $$