Simple expression for the number of elements in a diagonal half of a matrix

Question

Consider the following matrix:

$\begin{bmatrix} 0 & 0 & 0 & 0 \\ I & 0 & 0 & 0 \\ A & I & 0 & 0 \\ A & A & I & 0 \\ A & A & A & I \end{bmatrix}$

I need an expression for the number elements $A$ in the half below the diagonal $I$. Here is one that can be easily derived:

$ N = (n-1) n - \sum_{i=1}^{n-1} i$

where $n$ denotes the number of columns.

However, I feel that this expression is not elegant at all. It's like I don't see the wood for all the trees. Can anyone propose a simpler expression?

Answer 1

Referring to the comments first of all, if $N$ only depends on $n$ we have: $$N=(n-1)\,n-\sum_{i=1}^{n-1}i=(n-1)\,n-\frac{(n-1)\,n}{2}=\frac{(n-1)\,n}{2},$$ this is $\frac{n^2-n}{2}$.