Tetrahedral number

A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number, Ten, is the sum of the first n triangular numbers, that is,

T e n = ∑ k = 1 n T k = ∑ k = 1 n k ( k + 1 ) 2 = ∑ k = 1 n ( ∑ i = 1 k i ) {\displaystyle Te_{n}=\sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}=\sum _{k=1}^{n}\left(\sum _{i=1}^{k}i\right)}

The tetrahedral numbers are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... (sequence A000292 in the OEIS)

Formula

The formula for the nth tetrahedral number is represented by the 3rd rising factorial of n divided by the factorial of 3:

T e n = ∑ k = 1 n T k = ∑ k = 1 n k ( k + 1 ) 2 = ∑ k = 1 n ( ∑ i = 1 k i ) = n ( n + 1 ) ( n + 2 ) 6 = n 3 ¯ 3 ! {\displaystyle Te_{n}=\sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}=\sum _{k=1}^{n}\left(\sum _{i=1}^{k}i\right)={\frac {n(n+1)(n+2)}{6}}={\frac {n^{\overline {3}}}{3!}}}

The tetrahedral numbers can also be represented as binomial coefficients:

T e n = ( n + 2 3 ) . {\displaystyle Te_{n}={\binom {n+2}{3}}.}

Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle.

Proofs of formula

This proof uses the fact that the nth triangular number is given by

T n = n ( n + 1 ) 2 . {\displaystyle T_{n}={\frac {n(n+1)}{2}}.}

It proceeds by induction.

Base case

T e 1 = 1 = 1 ⋅ 2 ⋅ 3 6 . {\displaystyle Te_{1}=1={\frac {1\cdot 2\cdot 3}{6}}.}

Inductive step

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