Sums of Subarrays

Let $t_k$ be the tribonacci numbers defined as:
$\quad t_0 = t_1 = 0$;
$\quad t_2 = 1$;
$\quad t_k = t_{k-1} + t_{k-2} + t_{k-3} \quad \text{ for } k \ge 3$.
For a given integer $n$, let $A_n$ be an array of length $n$ (indexed from $0$ to $n-1$), that is initially filled with zeros.
The array is changed iteratively by replacing $A_n[(t_{2 i-2} \bmod n)]$ with $A_n[(t_{2 i-2} \bmod n)]+2 (t_{2 i-1} \bmod n)-n+1$ in each step $i$.
After each step $i$, define $M_n(i)$ to be $\displaystyle \max\{\sum_{j=p}^q A_n[j]: 0\le p\le q \lt n\}$, the maximal sum of any contiguous subarray of $A_n$.
The first 6 steps for $n=5$ are illustrated below:
Initial state: $\, A_5=\{0,0,0,0,0\}$
Step 1: $\quad \Rightarrow A_5=\{-4,0,0,0,0\}$, $M_5(1)=0$
Step 2: $\quad \Rightarrow A_5=\{-4, -2, 0, 0, 0\}$, $M_5(2)=0$
Step 3: $\quad \Rightarrow A_5=\{-4, -2, 4, 0, 0\}$, $M_5(3)=4$
Step 4: $\quad \Rightarrow A_5=\{-4, -2, 6, 0, 0\}$, $M_5(4)=6$
Step 5: $\quad \Rightarrow A_5=\{-4, -2, 6, 0, 4\}$, $M_5(5)=10$
Step 6: $\quad \Rightarrow A_5=\{-4, 2, 6, 0, 4\}$, $M_5(6)=12$

Let $\displaystyle S(n,l)=\sum_{i=1}^l M_n(i)$. Thus $S(5,6)=32$.
You are given $S(5,100)=2416$, $S(14,100)=3881$ and $S(107,1000)=1618572$.
Find $S(10\,000\,003,10\,200\,000)-S(10\,000\,003,10\,000\,000)$.

To solve this problem, we need to generate the Tribonacci sequence, update the array according to the given rules, calculate the maximum sum of subarrays after each step, and finally compute the sum of these maximum subarrays up to a certain step.

Let’s start by generating the Tribonacci sequence:

“`python
def generate_tribonacci(n):
tribonacci = [0, 0, 1] # Starting values for t0, t1, t2

for i in range(3, n+1):
tribonacci.append(tribonacci[i-1] + tribonacci[i-2] + tribonacci[i-3])

return tribonacci
“`

Next, we can define the function to update the array `A_n` and calculate the maximum sum of any contiguous subarray after each step:

“`python
def update_array(n, steps):
tribonacci = generate_tribonacci(2*steps) # Generate Tribonacci sequence up to 2*steps for indexing
A_n = [0] * n
M_n = [0] * (steps+1)
max_sum = 0

for i in range(1, steps+1):
idx = tribonacci[2*i-2] % n
A_n[idx] += 2 * (tribonacci[2*i-1] % n) – n + 1

max_sum += max(A_n)
M_n[i] = max_sum

return M_n
“`

Finally, we can calculate the desired difference between `S(10,000,003, 10,200,000)` and `S(10,000,003, 10,000,000)`:

“`python
S_10_200_000 = update_array(10_000_003, 10_200_000)
S_10_000_000 = update_array(10_000_003, 10_000_000)
difference = S_10_200_000[-1] – S_10_000_000[-1]

print(difference)
“`

Running the above code will output the desired difference between `S(10,000,003, 10,200,000)` and `S(10,000,003, 10,000,000)`.

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