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# Tetrahedral numbers

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

${\displaystyle T_{n}=\sum _{i=1}^{n}t_{i},\,}$

with ${\displaystyle \scriptstyle t_{i}\,}$ being the ${\displaystyle \scriptstyle i\,}$th triangular number.

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

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

where ${\displaystyle \scriptstyle n^{(k)}\,}$ 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, ...}