# Definition:Triangular Number

## Definition

Triangular numbers are those denumerating a collection of objects which can be arranged in the form of an equilateral triangle.

They are often denoted:

$T_1, T_2, T_3, \ldots$

### Definition 1

$T_n = \begin{cases} 0 & : n = 0 \\ n + T_{n-1} & : n > 0 \end{cases}$

### Definition 2

$\ds T_n = \sum_{i \mathop = 1}^n i = 1 + 2 + \cdots + \paren {n - 1} + n$

### Definition 3

$\forall n \in \N: T_n = P \left({3, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({3, n - 1}\right) + \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

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:

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

## Also known as

Triangular numbers are also known as triangle numbers.

## Also see

• Results about triangular numbers can be found here.

## Historical Note

The triangular numbers were named thus by the ancient Greeks, who formed them by adding up the series:

$1 + 2 + 3 + 4 + 5 + \cdots$