Pentagonal numbers are a type of figurate number that represent a pentagon. They can be calculated using the formula P_n = \fracn(3n - 1)2 , where n is a positive integer. The first few pentagonal numbers are 1, 5, 12, 22, and 35. Each number corresponds to a specific arrangement of dots that can form a pentagon. These numbers are part of a broader category known as polygonal numbers, which includes triangular numbers and hexagonal numbers. Pentagonal numbers have applications in various areas of mathematics, including number theory and combinatorics, and they can also be found in certain patterns in nature and art.