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# Problem 477

Number Sequence Game

The number sequence game starts with a sequence S of N numbers written on a line.

Two players alternate turns. At his turn, a player must select and remove either the first or the last number remaining in the sequence.

The player score is the sum of all the numbers he has taken. Each player attempts to maximize his own sum.

If N = 4 and S = {1, 2, 10, 3}, then each player maximizes his score as follows:

• Player 1: removes the first number (1)
• Player 2: removes the last number from the remaining sequence (3)
• Player 1: removes the last number from the remaining sequence (10)
• Player 2: removes the remaining number (2)

Player 1 score is 1 + 10 = 11.

Let F(N) be the score of player 1 if both players follow the optimal strategy for the sequence S = {s1, s2, …, sN} defined as:

• s1 = 0
• si+1 = (si2 + 45) modulo 1 000 000 007

The sequence begins with S = {0, 45, 2070, 4284945, 753524550, 478107844, 894218625, ……}.

You are given F(2) = 45, F(4) = 4284990, F(100) = 26365463243, F(104) = 2495838522951.

Find F(108).

• 玩家1：擦去第一个数（1）
• 玩家2：擦去剩余数列的最后一个数（3）
• 玩家1：擦去剩余数列的最后一个数（10）
• 玩家2：擦去最后剩下的数（2）

• s1 = 0
• si+1 = (si2 + 45) modulo 1 000 000 007