# 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).