Triangular numbers

A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The $n$ th triangular number is the number of dots composing a triangle with $n$ dots on a side, and is equal to the sum of the $n$ natural numbers from 1 to $n$ . The sequence of triangular numbers (sequence in the OEIS), starting at the 0th triangular number, is

The triangle numbers are given by the following explicit formulas:

where $\textstyle {n+1 \choose 2}$ is a binomial coefficient. It represents the number of distinct pairs that can be selected from $n + 1$ objects, and it is read aloud as " $n$ plus one choose two".

Carl Friedrich Gauss is said to have found this relationship in his early youth, by multiplying $n / 2$ pairs of numbers in the sum by the values of each pair $n + 1$ . 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.

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