Wiktionary
n. (context mathematics English) Any integer that is the sum of ''n'' natural numbers from 1 to ''n''.
Wikipedia
A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The th triangular number is the number of dots composing a triangle with dots on a side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers , starting at the 0th triangular number, is
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406 …The triangle numbers are given by the following explicit formulas:
T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +n = \frac{n(n+1)}{2} = {n+1 \choose 2} ,
where $\textstyle {n+1 \choose 2}$ is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two".
Carl Friedrich Gauss is said to have found this relationship in his early youth, by multiplying pairs of numbers in the sum by the values of each pair . However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus.
The triangular number solves the "handshake problem" of counting the number of handshakes if each person in a room with people shakes hands once with each person. In other words, the solution to the handshake problem of people is . The function is the additive analog of the factorial function, which is the products of integers from 1 to .
The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation:
L_n = 3 T_{n-1}= 3{n \choose 2};~~~L_n = L_{n-1} + 3(n-1), ~L_1 = 0.
In the limit, the ratio between the two numbers, dots and line segments is
\lim_{n\to\infty} \frac{T_n}{L_n} = \frac{1}{3}.
Usage examples of "triangular number".
He knew that he even had a degree of rarity: he was one of the few numbers to be both a square and a triangular number.