Ordinary Generating functions for $b_n$ Problem
Let $f(x)$ be a ordinary generating function for the sequence $ \{\ a_0, a_1, a_2... \}\ $ Find the ordinary generating function for $b_0 = b_1 = 0, b_2 = 1$ $b_n = a_n$ for $n \geq 3$.
Also find the generating function for $b_n = 0$ for even $n$, $b_n = a_n$ for odd $n$.

My attempts
For the first problem, $A(x) = 0+0+1x^2+a_3x^3+a_4x^4...$ 
So $$g(x) = x^2 + \sum_{n=3}^\infty a_n x^n$$
$$\Rightarrow x^2 + \sum_{n=0}^\infty a_n x^{n+3}$$
$$ \Rightarrow x^2 + \sum_{n=0}^\infty a_n x^n \cdot x^{-3}$$
$$\Rightarrow x^2 + f(x) \cdot x^{-3}$$
For the next problem I've got the sequence to be: $$ \{\ 0,a_1 , 0 , a_2 , 0, a_3\ldots \}\ $$
So $$A(x) = \{\ a_1 x + a_3 x^3 + a_5 x^5 \ldots \}\ $$
Which implies $$g(x) = \sum_{n=0}^\infty a_n x^n - \sum_{n=0}^\infty a_{2n} x^n$$
$$g(x) = f(x) - \sum_{n=0}^\infty a_{2n} x^n$$
But I can't get further. I hope my progress so far is correct in these.
Thank you.

So for the first one there are a couple of issues, when you go from summing from $n = 3$ to $n=0$ you add 3 on to the power, but not the index of the coefficient.
Also when you factor out the $x^3$ it turns into $x^{-3}$. Fix these things and see how far you can get.
For the second I'd recommend looking at $f(-x)$ and comparing that to $f(x)$.
