Notation question (factoral squares?)

Question

Not sure whether factoral is even the right word to use in this context but I'm looking for how you write 'the sum of all the squares' eg

3 = 14 (3^2 + 2^2 + 1^2)

4 = 30 (4^2 + 3^2 + 2^2 + 1^2)

Answer 1

The sum of squares actually has a closed form: $$ 1^2 + 2^2 + \cdots + n^2 = \sum_{i=1}^n i^2 = \frac{n(n+1/2)(n+1)}{3}. $$ Because the sum is closed, there is no designated notation for it. Something like the factorial function, on the other hand, does have a designated notation because it does not have any simple closed form. The factorial function also comes up more often than the sum of squares does.

Answer 2

Yes, these are called pyramidal numbers, or sometimes "square pyramidal" numbers. You can think of the "squares" stacked on top of each other, getting smaller and smaller to the apex like an ancient Egyptian monument.