I have an assignment problem that i have been fighting with for a while now..
I have this recursive function:
$a_n=\begin{cases} 3,&\text{if }n=0\\ 5,&\text{if }n=1\\ 4a_{n-1}-4a_{n-2},&\text{if }n\ge 2\;. \end{cases}$
We define the generating function $P(n)=\sum_{n=0}^\infty (a_n+a_{n+1})x^n$
Now I need to use the definition of $a_n$ in the recursive function to show that $P(n)=\frac{8-19x}{(1-2x)^2}$
I cant really get to this result, and I have been trying all sorts of things by now, nothing really leading me anywhere..
I hope some of you can help me! Thanks