How to denote a FOR...LOOP in mathematics

I am trying to help my son with a fairly basic maths question for school.

How many squares are on a standard $8times8$ chess board?

We have solved this one quickly, and then explored a number of different ways to calculate the answer in a number of different languages, all similar to the below:

$ text{Size} = 8 $
$ text{Squares} = text{FOR } 1 text{ TO Size: Squares} = text{Squares} + text{Size}^2 $

Is there a way to express this kind of loop as a mathematic equation? We’ve searched some texts at home and online, and got to this

$N=sum_{s=1}^n s^2$

but I’m not confident this is correct.

I’m trying to count all of the squares of any size. Where $S = 8$ (the size of a chessboard) $N$ (the number of any size square) $= (8 times 8)+(7 times 7)+(6 times 6)+(5 times 5)+(4 times 4)+(3 times 3)+(2 times 2)+(1 times 1) = 204 $.