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Problem 948

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Left vs Right

Left and Right play a game with a word consisting of L’s and R’s, alternating turns. On Left’s turn, Left can remove any positive number of letters, but not all the letters, from the left side of the word. Right does the same on Right’s turn except that Right removes letters from the right side. The game continues until only one letter remains: if it is an ‘L’ then Left wins; if it is an ‘R’ then Right wins.

Let $F(n)$ be the number of words of length $n$ where the player moving first, whether it’s Left or Right, will win the game if both play optimally.

You are given $F(3)=4$ and $F(8)=181$.

Find $F(60)$.

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小左对小右（一）

小左和小右在用一个由字母L和R组成的单词玩游戏，两人轮流行动。在小左的回合中，小左可以从单词的左侧移除任意正数个字母，但不能移除所有字母；小右则类似地从单词的右侧移除字母。游戏持续进行，直到只剩下一个字母：如果该字母是L则小左获胜；如果是R则小右获胜。

记$F(n)$为长度为$n$且在双方均采取最优策略时先手玩家（无论是小左还是小右）必胜的单词数量。

已知$F(3)=4$，$F(8)=181$。

求$F(60)$。

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