# 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.

### Examples of Triangular Numbers

The first few triangular numbers are as follows:

### Sequence of Triangular Numbers

The sequence of triangular numbers, for $n \in \Z_{\ge 0}$, begins:

$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, \ldots$