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Tetrahedral numbers or
triangular pyramidal numbers are 3dimensional
figurate numbers representing
tetrahedra (or tetrahedrons). The
th tetrahedral number is given by the formula

with
being the
th triangular number.
Tetrahedral numbers can also be obtained from binomial coefficients (which means that they can be looked up in Pascal’s triangle)

Tn = ( ^{n + 3}_{3} ) − ( ^{n + 2}_{2} ) = ( ^{n + 2}_{3} ) = , 
where
is the
rising factorial.
A000292 Tetrahedral (or triangular pyramidal) numbers:
a (n) = ( ^{n + 2}_{3} ) = . 

{0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, ...}
See also