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 $T_1, T_2, T_3, \ldots$, and they are formally defined as:
 * $\displaystyle T_n = \sum_{i \mathop = 1}^n i = 1 + 2 + \cdots + \left({n-1}\right) + n$

Specific Values of Triangular Numbers
From the definition:
 * $T_0 = 0$



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: $T_5 = 1 + 2 + 3 + 4 + 5 = 15$.

Also known as
Triangular numbers are also known as triangle numbers.

Also see
0 & : n = 0 \\ n + T_{n-1} & : n > 0 \end{cases}$
 * Recurrence Relation for Triangular Numbers: $T_n = \begin{cases}


 * Closed Form for Triangular Numbers: $T_n = \dfrac {n \left({n + 1}\right)} 2$

Historical Note
The triangular numbers were classified and investigated by the Pythagorean school in the $6$th century BCE. This was possibly the first time this had ever been done.