tetrahedral number

An integer of the form

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

where $n$ is a nonnegative integer. Sometimes referred to as $T_n$, tetrahedral numbers are listed in A000292 of Sloane’s OEIS. $2|T_n$ except when $n\equiv 1mod4$.

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

$$T_n=\left(\genfrac{}{}{0pt}{}{n+2}{3}\right).$$

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

Tetrahedral numbers have practical applications in sphere packing.