Generating function of a Fibonacci series but with certain variation.

Question

Let $f_n$ denote the $nth$ Fibonacci number then what is the generating function for the sequence$f_0,0,f_2,0,f_4,0,...$ $$\text{Attempt}$$Its known that the sequence has following two properties.\begin{align} \smash[b]{\sum_{i=1}^n F_{2i-1}}&=F_1+F_3+F_5+\cdots+F_{2n-1}\\ &=F_{2n}\\ \end{align} $$\begin{align} \smash[b]{\sum_{i=1}^n F_{2i}}&=F_2+F_4+F_6+\cdots+F_{2n}\\ &=F_{2n+1}-1\\ \end{align}$$ I will intentionally not start from $0$. $$B(x)=F_2x^1+0+F_4x^2+..=\sum_{k=1}^{\infty}F_{2k}x^k \implies B(1)=\sum_{k=1}^{\infty}\sum_{i=1}^{k}F_{2i-1}$$.But i dont see how to proceed from here. Any help will be appreciated!

Answer 1

Hint. If $f$ is the generating function of any sequence $f_0,f_1,f_2,f_3,f_4\dots$ then $$\frac{f(x)+f(-x)}{2}=f_0+f_2x^2+f_4x^4+\dots.$$