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Problem 364

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Comfortable distance

There are N seats in a row. N people come after each other to fill the seats according to the following rules:

1. If there is any seat whose adjacent seat(s) are not occupied take such a seat.
2. If there is no such seat and there is any seat for which only one adjacent seat is occupied take such a seat.
3. Otherwise take one of the remaining available seats.

Let T(N) be the number of possibilities that N seats are occupied by N people with the given rules. The following figure shows T(4)=8.

We can verify that T(10) = 61632 and T(1 000) mod 100 000 007 = 47255094.

Find T(1 000 000) mod 100 000 007.

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舒适距离

有N个座位排成一行。N个人陆续前来，根据以下规则选择座位：

1. 如果存在空座位两侧都没有人，则选择这样的座位。
2. 如果没有这样的座位，但存在空座位只有一侧有人，则选择这样的座位。
3. 否则，选择任意一张剩下的空座位。

记T(N)为在上述规则下N个人坐N个座位的所有可能性。下图说明T(4)=8。

我们可以验证T(10) = 61632以及T(1 000) mod 100 000 007 = 47255094。

求T(1 000 000) mod 100 000 007。

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