tetrahedral number

A tetrahedral number is a figurate number that represents the count of objects arranged in the shape of a pyramid with a triangular base and three sides, like a pyramid made of spheres. It is the sum of the first n triangular numbers, forming a three-dimensional shape resembling a tetrahedron. For example, the 3rd tetrahedral number (4) counts the total spheres in a pyramid with four layers: 1 in the top layer, then 3 below, and 6 at the bottom, totaling 10. Tetrahedral numbers are useful in combinatorics and geometry, describing arrangements and volumes of certain 3D patterns.