859. Cookie Game

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

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

Find $C(300)$$C(300)$.

859. 曲奇游戏

小奇和小偶正用 $N$$N$ 片曲奇玩游戏。游戏开始前，这 $N$$N$ 片曲奇将被分为若干大小不一定相等的堆。

随后，由小奇先手，二人轮流进行操作：

• 在小奇的回合中，他可以选择含奇数片曲奇的一堆曲奇，吃掉其中一片曲奇后，将该堆中剩余的曲奇均分成两堆。

• 在小偶的回合中，他可以选择含偶数片曲奇的一堆曲奇，吃掉其中两片曲奇后，将该堆中剩余的曲奇均分成两堆。

无法行动的玩家落败。

记：将 $N$$N$ 片曲奇分堆的所有方法中，有 $C(N)$$C(N)$ 种能使得小偶有必胜策略。例如，$C(5)=2$$C(5) = 2$。因为你可以将 $5$$5$ 片曲奇单独成堆，或者将其分为大小分别为 $2$$2$、$2$$2$、$1$$1$ 的三堆，这两种情况下，小偶都有必胜策略。

已知 $C(16)=64$$C(16) = 64$。

求 $C(300)$$C(300)$。

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