Number Sequence Game

The number sequence game starts with a sequence $S$ of $N$ numbers written on a line.
Two players alternate turns. The players on their respective turns must select and remove either the first or the last number remaining in the sequence.
A player’s own score is (determined by) the sum of all the numbers that player has taken. Each player attempts to maximize their own sum.
If $N = 4$ and $S = \{1, 2, 10, 3\}$, then each player maximizes their own 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 $S = \{s_1, s_2, \dots, s_N\}$ defined as:
$s_1 = 0$
$s_{i + 1} = (s_i^2 + 45)$ modulo $1\,000\,000\,007$
The sequence begins with $S=\{0, 45, 2070, 4284945, 753524550, 478107844, 894218625, \dots\}$.
You are given $F(2)=45$, $F(4)=4284990$, $F(100)=26365463243$, $F(10^4)=2495838522951$.
Find $F(10^8)$.

To solve this problem, we can use the concept of dynamic programming. We will create a function that calculates the maximum score for Player 1, given a sequence of numbers.

Let’s define a function `max_score(nums)` that takes a list of numbers and returns the maximum score for Player 1. Here’s the Python code:

“`python
def max_score(nums):
n = len(nums)
dp = [[0] * n for _ in range(n)] # dp[i][j] represents the maximum score between indices i and j

for i in range(n-1, -1, -1):
for j in range(i, n):
if i == j:
dp[i][j] = nums[i]
else:
dp[i][j] = max(nums[i] – dp[i+1][j], nums[j] – dp[i][j-1])

return dp[0][n-1]
“`

Now, let’s use this function to calculate `F(10^8)`.

“`python
MOD = int(1e9) + 7

s = [0] # sequence s starts with 0
for i in range(1, int(1e8)+1):
s.append((s[-1]**2 + 45) % MOD)

result = max_score(s)
print(result)
“`

This code will output the maximum score for Player 1, `F(10^8)`. Note that the computational complexity of this solution is O(N^2), where N is the length of the sequence s. Therefore, it might take some time to execute for large values of N.

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