Tetrahedral numbers or triangular pyramid numbers are the numbers that can be arranged into a pyramid with a triangular base (i.e. a tetrahedron). Each layer of a tetrahedral number is a triangular number, and its formula is the sum of the first n triangular numbers, i.e. n(n + 1)(n + 2) / 6. Its first few terms are: 1, 4, 10, 20, 35, 56, 84, 120... (OEIS: A000292)

The even-odd arrangement of tetrahedral numbers is "odd-even-even-even".

In 1878, AJ Meyl proved that there are only three tetrahedral numbers that are also square numbers: 1, 4, and 19600. The only number that is both a tetrahedral number and a square pyramid number is 1 (Beukers (1988)).

They can be found in the 4th item in each row of the Pascal's triangle from right to left or from left to right.

formula

Tetrahedron number or triangular pyramid number (T_n) = ( n × (n+1) × (n+2) ) / 6