Generating Function of Even Fibonacci

Question

I was posed the following question recently on an exam:

Determine the generating function of the even-indexed Fibonacci numbers $F_{2n}$ given that the generating function of Fibonacci numbers is $\frac{x}{1-x-x^2}$.

I could not think of how apply knowledge of the generating function of the Fibonacci numbers to determine the generating function of the even Fibonacci, so I tried to to determine the generating function using the identity $F_{2n}=F_{n+1}^2-F_{n-1}^2$, but to no avail.

Any thoughts about how to find its generating function?

Answer 1

HINT: For $f(x) = \displaystyle\sum_{n \geqslant 0} c_n x^n$, what is the series for $\frac{1}{2} \left( f(x) + f(-x)\right)$?