Tetrahedral number

A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a base and three sides, that is, a tetrahedron. The tetrahedral number for n is the sum of the first n triangular numbers added up.

The first few tetrahedral numbers are

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969

Tetrahedral numbers can be modelled in physical space. For example, the tetrahedral number 35 can be modelled with 35 billiard balls and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.

The parity of tetrahedral numbers follows the pattern odd-even-even-even.

In 1878, A.J. Meyl proved that there are only three tetrahedral numbers that are also square, namely, 1, 4 and 19600. So far, the only known tetrahedral number that is also a square pyramidal number is 1.

Tetrahedral numbers are found in the fourth position either from left to right or right to left in Pascal's triangle.