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Comfortable distance

Problem 364

Published on Saturday, 24th December 2011, 01:00 pm; Solved by 365

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.