Odd and Even are playing a game with $N$ cookies.

The game begins with the $N$ cookies divided into one or more piles, not necessarily of the same size. They then make moves in turn, starting with Odd.
Odd's turn: Odd may choose any pile with an odd number of cookies, eat one and divide the remaining (if any) into two equal piles.
Even's turn: Even may choose any pile with an even number of cookies, eat two of them and divide the remaining (if any) into two equal piles.
The player that does not have a valid move loses the game.

Let $C(N)$ be the number of ways that $N$ cookies can be divided so that Even has a winning strategy.
For example, $C(5)=2$ because there are two winning configurations for Even: a single pile containing all five cookies; three piles containing one, two and two cookies.
You are also given $C(16)=64$.

Find $C(300)$.