Triangular number

A triangular number is a number of objects that can be arranged in an equilateral triangular pattern. The nth triangular number is the sum of the integers from 1 through n.

Arithmetic progression: A sequence whose successive terms differ by a fixed amount. The rows of a triangular arrangement form an arithmetic progression: 1, 2, 3, and so on.
Natural number: A positive integer used for counting; some conventions also include zero. The index n and the row sizes are natural numbers.
Handshake problem: A counting problem asking how many pairwise handshakes occur among a group of people. Each pair contributes one handshake, giving the triangular count for the group size.
Square number: An integer that is the product of an integer with itself. Square numbers form square arrays, unlike the triangular arrangements defining this sequence.
Triangular number formula: The formula giving the nth triangular number as n(n + 1)/2. Pairing terms in 1 + 2 + … + n derives this compact expression.
Summation: The operation of adding a sequence of terms, often written with sigma notation. A triangular number is defined by summing consecutive positive integers.
Triangular array: A data arrangement whose rows contain successively increasing numbers of entries. Counting all entries in rows of lengths 1 through n produces a triangular number.
Tetrahedral number: A number counting objects arranged in a tetrahedral stack of triangular layers. It extends triangular layering into three dimensions by summing triangular numbers.
Binomial coefficient: A number counting ways to choose a fixed-size subset from a finite set. The nth triangular number equals the number of ways to choose two objects from n + 1.
Integer: A whole number that may be positive, negative, or zero. Every triangular number is an integer, and its formula uses integer arithmetic.

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