Editing Tetrahedral number (section)

{{Short description|Polyhedral number representing a tetrahedron}}
[[Image:Pyramid of 35 spheres animation.gif|frame|right|A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.]]
A '''tetrahedral number''', or '''triangular pyramidal number''', is a [[figurate number]] that represents a [[pyramid (geometry)|pyramid]] with a triangular base and three sides, called a [[tetrahedron]]. The {{mvar|n}}th tetrahedral number, {{mvar|Te<sub>n</sub>}}, is the sum of the first {{mvar|n}} [[triangular number]]s, that is,

:<math> Te_n = \sum_{k=1}^n T_k = \sum_{k=1}^n \frac{k(k+1)}{2} = \sum_{k=1}^n \left(\sum_{i=1}^k i\right)</math>

The tetrahedral numbers are:

:[[1]], [[4]], [[10]], [[20 (number)|20]], [[35 (number)|35]], [[56 (number)|56]], [[84 (number)|84]], [[120 (number)|120]], [[165 (number)|165]], [[220 (number)|220]], ... {{OEIS|id=A000292}}