Definition:Triangular Number

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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

$\forall n \in \N: T_n = \map P {3, n} = \begin{cases} 0 & : n = 0 \\ \map P {3, n - 1} + \paren {n - 1} + 1 & : n > 0 \end{cases}$

where $\map P {k, n}$ denotes the $k$-gonal numbers.


Examples of Triangular Numbers

The first few triangular numbers are as follows:

TriangleNumbers.png


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.

Some sources refer to a triangular number just as a triangular.


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:

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


Sources