Lets say we are arranging a stack of spheres in this manner:
To place a sphere over other spheres one needs atleast 3 spheres below it to support it . When one brings 3 spheres together one can put another sphere above the gap formed by the three spheres.
How could we find the number of spheres in $N$ layers?
This problem is similar to finding the number of elements in a two-dimensional pyramid, where each successive row downwards of the pyramid has one more element than the one above (basic summation). Then, the number of elements in the $N^{th}$ layer is $\sum_{i=1}^N i$
To find the total number of spheres, we need to sum all the top $N$ layers. Then, we have $\sum_{j=1}^N \sum_{i=1}^j i$