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Problem 859

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Cookie Game

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 $C(300)$.

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饼干游戏

小奇和小偶在玩一个游戏，需要用到$N$块饼干。

游戏开始时，将$N$块饼干分成若干堆，每一堆的数量不必相同。两人轮流行动，由小奇先开始。

轮到小奇时：小奇可以任选一堆有奇数块的饼干，吃掉一块，并将剩下的饼干（若有）分成相等的两堆。
轮到小偶时：小偶可以任选一堆有偶数块的饼干，吃掉两块，并将剩下的饼干（若有）分成相等的两堆。
首先无法行动的玩家输掉游戏。

记$C(N)$为小偶有必胜策略的初始局面的数目。

例如，$C(5) = 2$，对应的两种初始局面是：所有五块饼干划分为一堆，或者划分为三堆，分别包含一、二、二块饼干。

已知$C(16) = 64$。

求$C(300)$。

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