Nontriangular

Nontriangular can refer to two related but distinct ideas in mathematics. In geometry, it describes a polygon that is not a triangle, such as a quadrilateral or pentagon. In number theory, it refers to natural numbers that are not triangular numbers. The two uses share a naming origin but describe different objects: shapes in geometry versus a subset of the positive integers in arithmetic.

Triangular numbers are the positive integers that can form an equilateral triangular dot arrangement. They are given by the formula T_n = n(n+1)/2 for n ≥ 1, producing the sequence 1, 3, 6, 10, 15, 21, and so on (OEIS sequence A000217). The nth triangular number counts the number of items in a triangular array with n items on each side.

A nontriangular number is any positive integer that is not triangular. A useful way to recognize triangular numbers is to check whether 8m+1 is a perfect square: a number m is triangular if and only if 8m+1 equals k^2 for some integer k (which will be odd). Consequently, nontriangular numbers are those m for which 8m+1 is not a perfect square. Examples of nontriangular numbers include 2, 4, 5, 7, 8, 9, 11, 12, 13, 14, and so on.

Counting nontriangular numbers up to a bound N is given by N minus the count of triangular numbers up to N, which is floor((sqrt(8N+1) − 1)/2). Therefore, the number of nontriangular numbers ≤ N is N − floor((sqrt(8N+1) − 1)/2). The density of nontriangular numbers among the positive integers tends to 1 as N grows, since triangular numbers have asymptotic density zero.

In geometry, nontriangular simply denotes polygons with at least four sides, contrasting with triangles. The term is less common in rigorous geometric terminology, where more specific descriptors (quadrilateral, pentagon, etc.) are usually preferred.