Finding a generating function for a sequence with two recurrence equations

Question

The sequence $a_{n}$ is defined as follows:

1. $a_{0}$ = 0 , $a_{1}$ = 1
2. $a_{2n} = a_{n}$
3. $a_{2n+1} = a_{n} + a_{n+1}$

let the generating function $F(x)$ be defined as $F(x) = \sum_{n=1}^{\infty} a_{n}x^{n-1}$

I'm curious as to how I would find $F(x)$.

In my attempt, I rearranged terms so that $a_{2n+1} = a_{2n} + a_{2n+2}$ then defined $j=2n$ and rewrote it as $a_{j+2} = a_{j} - a_{j+1}$. I then multiplied by $x^j$ across and used sums to find the answer, which was wrong. Am I going about this the wrong way, or is there an easier method?

Thanks