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?