Tetrahedral numbers

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Tetrahedral numbers or triangular pyramidal numbers are 3-dimensional figurate numbers representing tetrahedra (or tetrahedrons). The

n

th tetrahedral number is given by the formula

Tn  =    n ∑   i  = 1    ti  ,

with

ti

being the

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)

Tn  =  (  n + 33  )  − (  n + 22  )   =  (  n + 23  )   =  n ( 3 ) 3!  ,

where

n (k )

is the rising factorial.
A000292 Tetrahedral (or triangular pyramidal) numbers:

a (n) = (  n + 23  )  = 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

• 4-faced Platonic numbers.
• 3-dimensional simplicial polytopic numbers.