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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

2 Answers 2