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Tetrahedral numbers

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

T_n = \sum_{i = 1}^{n} t_i, \,

with \scriptstyle t_i \, being the \scriptstyle i \,th triangular number.

Tetrahedral numbers can also be obtained from binomial coefficients (which means that they can be looked up in Pascal's triangle)

T_n = \binom{n + 3}{3} - \binom{n + 2}{2} = \binom{n + 2}{3} = \frac{n^{(3)}}{3!}, \,

where \scriptstyle n^{(k)} \, is the rising factorial.

A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.

{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

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