The following game is a classic example of Combinatorial Game Theory:
Two players start with a strip of n white squares and they take alternate turns.
On each turn, a player picks two contiguous white squares and paints them black.
The first player who cannot make a move loses.
- If n = 1, there are no valid moves, so the first player loses automatically.
- If n = 2, there is only one valid move, after which the second player loses.
- If n = 3, there are two valid moves, but both leave a situation where the second player loses.
- If n = 4, there are three valid moves for the first player; she can win the game by painting the two middle squares.
- If n = 5, there are four valid moves for the first player (shown below in red); but no matter what she does, the second player (blue) wins.
So, for 1 ≤ n ≤ 5, there are 3 values of n for which the first player can force a win.
Similarly, for 1 ≤ n ≤ 50, there are 40 values of n for which the first player can force a win.
For 1 ≤ n ≤ 1 000 000, how many values of n are there for which the first player can force a win?
- 如果n = 1，不存在合法操作，先手玩家自动输掉游戏。
- 如果n = 2，只有一种合法操作，在此之后后手玩家输掉游戏。
- 如果n = 3，有两种合法操作，但在此之后后手玩家均输掉游戏。
- 如果n = 4，有三种合法操作，先手玩家只需将中间两个白格涂成黑色即可赢得游戏。
- 如果n = 5，有四种合法操作（如下图红色所示）；但无论先手玩家如何操作，后手玩家（蓝色所示）都将赢得游戏。
因此，对于1 ≤ n ≤ 5，共有3种n值使得先手玩家必胜。
同样地，对于1 ≤ n ≤ 50，共有40种n值使得先手玩家必胜。
对于1 ≤ n ≤ 1 000 000，共有多少种n值使得先手玩家必胜？