generating function Homework Question 1

Question

This is a HW question

I am asked to find a closed form generating function for $1,1,0,1,1,0,1,1,0....$ so then $f(x)=x^0+x^1+0x^2+x^3+x^4+0x^5+x^6+x^7+0x^8$

could use some hint or help.

Answer 1

Hint: See if writing it as follows helps:

$$f(x) = 1 + x + x^{3} +x^{4} + \cdots = ( 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + \cdots ) - (x^{2} + x^{5} + x^{8} + \cdots) = \frac{1}{1-x} - x^{2}(1+x^{3}+x^{6}+\cdots)$$

Can you take it from here?

Answer 2

Hint: You can view it as a sum of two simple infinite geometric series.