tetrahedral numbers

Only three tetrahedral numbers are also perfect squares.

The tetrahedral numbers can also be represented as binomial coefficients:

Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle.

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

Pentatopal numbers can also be represented as the sum of the first n tetrahedral numbers.

Square pyramidal numbers are also related to tetrahedral numbers in a different way:

Tetrahedral numbers can be modelled by stacking spheres.

The first few tetrahedral numbers are:

The infinite sum of tetrahedral numbers reciprocals is 3/2, which can be derived using telescoping series:

A. J. Meyl proved in 1878 that only three tetrahedral numbers are also perfect squares, namely:

An observation of tetrahedral numbers:

The next pair of diagonals contain the tetrahedral numbers in order, and the next pair give pentatope numbers.

In a use going back to Jakob Bernoulli's Ars Conjectandi, the term figurate number is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successive triangular numbers, etc.

The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers.

Another relation between octahedral numbers and tetrahedral numbers is also possible, based on the fact that an octahedron may be divided into four tetrahedra each having two adjacent original faces (or alternatively, based on the fact that each square pyramidal number is the sum of two tetrahedral numbers):