A tetrahedral number counts the points needed to build a tetrahedron with \(k\) points per edge: \(T_k = k(k+1)(k+2)/6\). The first: 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455.
Tetrahedral numbers are the 3-dimensional analog of triangular numbers. They are also the partial sums of triangular numbers: \(T_k = \sum_{i=1}^{k} \binom{i+1}{2}\).
A classic problem (the cannonball problem) asks: which tetrahedral numbers are also perfect squares? Only three: 1, 4, and 19600. The proof of this finite list, due to Mordell in 1969, is a striking result of Diophantine geometry.