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 otherwise called triangle numbers.

Or we can just say that a number is triangular.

They are often denoted $T_1, T_2, T_3, \ldots$, and they are formally defined as:

$\displaystyle T_n = \sum_{i=1}^n i = 1 + 2 + \cdots + \left({n-1}\right) + n$.

Thus $T_0 = 0$.

The first triangular number: $T_1 = 1$.

The second triangular number: $T_2 = 1 + 2 = 3$.

The third triangular number: $T_3 = 1 + 2 + 3 = 6$.

The fourth triangular number: $T_4 = 1 + 2 + 3 + 4 = 10$.

The fifth triangular number: $T_5 = 1 + 2 + 3 + 4 + 5 = 15$.

Recurrence Formula

It can be seen directly from the above that:

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

Closed Form

From Closed Form for Triangular Numbers, we have:

$T_n = \dfrac {n \left({n + 1}\right)} 2$

Also see

This sequence is A000217 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).