Definition:Triangular Number
Jump to navigation
Jump to search
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:
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
- Closed Form for Triangular Numbers: $T_n = \dfrac {n \paren {n + 1} } 2$
- 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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): figurate numbers
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.2$: Pythagoras (ca. $\text {580}$ – $\text {500}$ B.C.)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): triangular number