Neartriangular

A neartriangular number is a type of figurate number that approximates the properties of triangular numbers but does not follow the exact same sequence. Triangular numbers are generated by the formula for the nth triangular number, Tₙ = n(n + 1)/2, which represents a triangular arrangement of dots. Neartriangular numbers, however, are derived from a modified formula that introduces a small adjustment to this sequence.

The general formula for neartriangular numbers is often expressed as Tₙ' = n(n + k)/2, where k is a small integer (typically k = 3 or k = 5). This adjustment shifts the sequence slightly, creating numbers that are close to triangular numbers but not identical. For example, when k = 3, the sequence begins 1, 4, 9, 15, 22, 30, 39, 49, 60, 72, and so on. These numbers can be visualized as nearly triangular arrangements of dots, with minor deviations from the perfect triangular shape.

Neartriangular numbers are sometimes explored in recreational mathematics and combinatorics, where variations on standard figurate numbers provide interesting patterns and applications. They can also appear in problems involving tiling, lattice paths, or other geometric configurations where slight deviations from symmetry are permissible. While not as commonly studied as triangular or other well-known figurate numbers, they offer a useful framework for understanding how small modifications to a sequence can yield distinct yet related mathematical structures.