Definition:Triangular Number/Definition 3

Definition

Triangular numbers are defined as the sequence:

$\forall n \in \N: T_n = \map P {3, n} = \begin{cases}

0 & : n = 0 \\ \map P {3, n - 1} + \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ 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:

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

Also see

• Equivalence of Definitions of Triangular Number