figurate number

Figurate numbers are a class of numbers that can be represented by arrangements of dots forming geometric figures. These numbers were of particular interest to ancient Greek mathematicians, such as the Pythagoreans, who believed numbers possessed shapes.1

Examples of figurate numbers include:
* Triangular numbers: Represented by dots arranged in triangles, such as 1, 3, 6, 10, 15, etc. The formula for the nth triangular number is ( \frac{n(n+1)}{2} ).23
* Square numbers: Formed by square arrays of dots, like 1, 4, 9, 16, 25, etc..1
* Oblong numbers: Created by rectangular arrays where rows have one more dot than columns, such as 2, 6, 12, 20, 30.1

These numbers have fascinated mathematicians for centuries, with relationships discovered between their geometric patterns and algebraic expressions.1