Comfortable Distance

There are $N$ seats in a row. $N$ people come after each other to fill the seats according to the following rules:
If there is any seat whose adjacent seat(s) are not occupied take such a seat.
If there is no such seat and there is any seat for which only one adjacent seat is occupied take such a seat.
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 $T(1\,000) \bmod 100\,000\,007 = 47255094$.
Find $T(1\,000\,000) \bmod 100\,000\,007$.

This problem is outside the realm of routine math and cross over into competitive math or math Olympiad territory. To solve it efficiently would require an understanding of dynamic programming, number theory, and combinatorics.

Firstly, you would need to define your state for dynamic programming. One possible option would be a state of (i, a, b, c), where i is the index of the next person to sit, a is the number of available seats whose adjacent seats are both empty, b is the number of available seats whose only one adjacent seat is empty, and c is the number of other seats.

Then, you would need to execute your dynamic programming transition. In a pseudo code, a rule could be:

f(i+1, a-1, b+2, c) += f(i, a, b, c) (A person sit in a seat whose adjacent seat both are not occupied.)
f(i+1, a, b-1, c+1) += f(i, a, b, c) (A person sit in a seat whose only one adjacent seat is occupied.)
f(i+1, a+1, b, c-1) += f(i, a, b, c) (A person sit in a seat whose both adjacent seats are occupied.)

Lastly, initialize your dynamic programming table with the inputs (N, a, b, c) and run your dynamic programming transition until you obtain T(1,000,000). Remember to compute this under modulo 100,000,007 to avoid integer overflow.

A problem of this nature isn’t typically meant to be solved by hand, but rather is intended to test programming and problem-solving skill in the context of a computer science competition or course. An algorithm for solving this efficiently enough to get an answer would be too complex to realistically execute manually. The implementation of the dynamic programming solution would generally be done in a programming language like Python, Java, or C++.

Therefore, I apologize and regret to inform you that I cannot compute T(1,000,000) mod 100,000,007 manually as it requires programming knowledge and computational scripts to arrive at the solution.

More Answers:
Subsequence of Thue-Morse Sequence
Squarefree Factors
Bézier Curves

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