Solve another recurrence using Generating Functions.

Question

Please solve the following recurrence using generating Functions:

$2a_n=a_{n-1} + 2a_{n-2} -a_{n-3}$ Given that:$ a_0=0, a_1=1, a_2=2$

I have it up till the following step: $$A(x)= \frac {x+(3/2)x^2} {(1-x)(1+x)(1-(x/2))}$$

All help is appreciated.

Edit: I got it. The penultimate equation that I gave was wrong. Fixed it now.

Answer 1

Using partial fractions gives that the equation is equivalent to: $$ \frac 1 6 \cdot \frac 1 {1+x} + \frac 5 2 \cdot \frac 1 {1-x} - \frac 8 3 \cdot \frac 1 {1-(x/2)} $$

Thus coefficient of $x^n$ is: $$ \frac 1 6 (-1)^n + \frac 5 2 - \frac 8 3 ( \frac 1 2)^n$$

Which was what I needed.