Definition:Triangular Number/Definition 3

Definition
Triangular numbers are defined as the sequence:
 * $\forall n \in \N: T_n = P \left({3, n}\right) = \begin{cases}

0 & : n = 0 \\ P \left({3, n - 1}\right) + \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.

Also see

 * Equivalence of Definitions of Triangular Number