I'm trying to find the generating function for this sequence:
$$0,0,3,0,9,17,33,65,129,257...$$
What I know so far:
$$0\cdot x^0 + 0\cdot x^1 + 3\cdot x^2 + 0 \cdot x^3$$
and
$$x^5(9+17x+33x^2+65x^3+129^4+257^5....$$
So what I have to do now Is just add the two, but I'm not 100% sure how to give a closed form answer for the second sequence. From what I can see it has an arithmetic progression of $8\cdot n$, so I've gotten this:
$$3x^2+x^5\sum_{n=1}^∞(9+8n)$$
What else do I need to do ? have I found the generating function?
The expression you are worried about can be rewritten as $$(1+x+x^2+x^3+x^4+\cdots)+(8+16x+32x^2+64x^3+\cdots).$$ Each is a geometric series, and you can write down the sum for each.