Tetrahedron number, triangular pyramid number calculator

Tetrahedral numbers or triangular pyramidal numbers are numbers that can be arranged to form 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, A.J. Meyl proved that only three tetrahedral numbers are also square numbers: 1, 4, 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 Pascal's triangle from right to left or left to right.

Formula

Number of tetrahedrons or triangular pyramids (T_n) = ( n × (n+1) × (n+2) ) / 6