tetrahedral numbers or triangular pyramidal numbers are the numbers that can be arranged into pyramids with triangular bases (i.e. tetrahedrons). each layer of tetrahedral numbers is a triangle number, and its formula is the sum of the first n triangle numbers, that is, n(n + 1)(n + 2) / 6. the first few items are: 1, 4, 10, 20, 35, 56, 84, 120...（oeis:a000292）

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

in 1878, aj meyl proved that only 3 tetrahedral numbers are simultaneously square numbers: 1, 4, and 19600. the only number that is both a tetrahedral number and a square pyramidal number is 1 (beukers (1988)).

they can be found in the 4th item of each row of the yang hui triangle from right to left or left to right.

formula

tetrahedral number or triangular pyramid number (t_n) = ( n × (n+1) × (n+2) ) / 6