tetrahedral number

An integer of the form

 ${{(n^{2}+n)(n+2)}\over 6},$

where $n$ is a nonnegative integer. Sometimes referred to as $T_{n}$, are listed in A000292 of Sloane’s OEIS. $2|T_{n}$ except when $n\equiv 1\mod 4$.

With $t_{n}$ the $n$th triangular number, the $n$th tetrahedral number can be calculated with this formula:

 $T_{n}=\sum_{i=1}^{n}t_{i}.$

Another way to calculate tetrahedral numbers is with the binomial coefficient

 $T_{n}={n+2\choose 3}.$

This means that tetrahedral numbers can be looked up in Pascal’s triangle.

Tetrahedral numbers have practical applications in sphere packing.

Title tetrahedral number TetrahedralNumber 2013-03-22 15:56:34 2013-03-22 15:56:34 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Definition msc 11A99 triangular pyramidal number