Problem statement: You are given an integer $$$n$$$. There are n sticks of length $$$1, 2, 3, ..., n$$$. You find to find out how many incongruent triangles can be formed by using three of the given sticks. Output the result modulo $$$10^{12} + 9$$$.

Constraint: $$$4 < n < 10^{12}$$$, where $$$n$$$ is a positive integer.

My idea: I analyzed some base cases. After that, I transformed the problem into an equation with a nested loop. After resolving the nested loop, I determined the number of triangles, denoted as $$$x$$$, to be,

I can't optimize it for the ceiling function. It is working on $$$O(n)$$$. But the constraint of the problem is: $$$4 < n < 10^{12}$$$. Therefore, we can't iterate over n. However, Can it be simplified? (Can we express this equation without using the summation sign?)

Thanks in advance for your assistance.