tetrahedral number

An integer of the form $${{(n^2 + n)(n + 2)} \over 6},$$ where $n$ is a nonnegative integer. Sometimes referred to as $T_n$ , tetrahedral numbers 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 $$T_n={n+2\choose3}.$$ This means that tetrahedral numbers can be looked up in Pascal's triangle.

Tetrahedral numbers have practical applications in sphere packing.