What is the time complexity of an iteration through all possible sequences of an array

Question

An algorithm that goes through all possible sequences of indexes inside an array.

Time complexity of a single loop and is linear and two nested loops is quadratic O(n^2). But what if another loop is nested and goes through all indexes separated between these two indexes? Does the time complexity rise to cubic O(n^3)? When N becomes very large it doesn't seem that there are enough iterations to consider the complexity cubic yet it seems to big to be quadratic O(n^2)

Here is the algorithm considering N = array length

for(int i=0; i < N; i++)
{
    for(int j=i; j < N; j++)
    {
        for(int start=i; start <= j; start++)
        {
          //statement
        }
    }
}

Here is a simple visual of the iterations when N=7(which goes on until i=7):

And so on..

Should we consider the time complexity here quadratic, cubic or as a different size complexity?

Answer 1

For the basic

for (int i = 0; i < N; i++) {
    for (int j = i; j < N; j++) {
        // something
    }
}

we execute something n * (n+1) / 2 times => O(n^2). As to why: it is the simplified form of
sum (sum 1 from y=x to n) from x=1 to n.

For your new case we have a similar formula:
sum (sum (sum 1 from z=x to y) from y=x to n) from x=1 to n. The result is n * (n + 1) * (n + 2) / 6 => O(n^3) => the time complexity is cubic.

The 1 in both formulas is where you enter the cost of something. This is in particular where you extend the formula further.

Note that all the indices may be off by one, I did not pay particular attention to < vs <=, etc.