I'd like to see if this concept has a name, so I will present my understanding of it (with some notation made up for readability) in the hope that someone will recognize it.
A triangle number 1+2+3+..+n can be calculated with a formula Δ(n)=n(n+1)/2 and be illustrated with a triangle figure:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
where n is a row and Δ(n) is the sum of all elements in the triangle up to and including that row.
I've had reasons to use a triangle of the following shape when solving a problem:
1 2 3 4 5 6 7 8 9 10 11 12
It follows a similar pattern, only that the columns expand one step further to the right only every two rows.
The sum can be calculated with ΔΔ(n)=Δ(Floor(n/2))+Δ(Ceiling(n/2)).
(This can be confirmed with the last number on the 5h row being 9).
The intuition being that my triangle is just two regular lower triangles weaved into each other, so there's two sums alternating in increased range every row.
Does this triangle shape have a name, and if so, what is it, and is this method the usual way of calculating what I've called ΔΔ(n)?