hexagonal number

A **hexagonal number** is a figurate number that represents a pattern of dots arranged in the shape of a hexagon. The nth hexagonal number is given by the formula:
[ H_n = 2n^2 – n ]
where ( n ) is a positive integer. These numbers appear frequently in mathematics, particularly in number theory and geometry, and have interesting properties related to polygonal numbers and combinatorics. Examples include 1, 6, 15, 28, and 45, corresponding to ( n = 1, 2, 3, 4, ) and 5. Hexagonal numbers can also be visualized as dots forming a large hexagon with equal side lengths, making them useful in various mathematical patterns and problems.