Generating function for the sequence $(0,0,0,1,2,...,2^{r-3},...)$

Question

Find the generating function for the sequence $(0,0,0,1,2,...,2^{r-3},...)$

I found this question in my notes. I know the generating function for the sequence $a_r$ is $\sum a_r x^r$ but I'm not sure how to apply it in this case.

EDIT: Removed the typo in the sequence.

Answer 1

Hint: $$x^3+2x^4+4x^5+8x^6+\cdots = x^3(1+2x+(2x)^2+(2x)^3+\cdots)$$

Do you know a closed form for the part in parentheses to the right?