Let $\displaystyle f_n$ be the sequence of Fibonacci numbers.

It's known that $\displaystyle \sum_{n \ge 0} f_n x^n = \dfrac{1}{1-x-x^2}$

What is the interpretation of this for $\displaystyle \sum_{n \ge 0} f_{2n} x^n \text{ and }\sum_{n \ge 0} f_{2n+1} x^n\text{ ?}$

How do we prove it?