Pentagonalzahl

Pentagonalzahl, also known as pentagonal number, is a figurate number that extends the concept of triangular and square numbers to a five-sided polygon. The nth pentagonal number represents the number of dots in a pattern of dots arranged in a pentagon with n dots on a side. The formula to calculate the nth pentagonal number is given by:

Pn = (3n^2 - n) / 2

This formula can be derived by observing the pattern of dots in a pentagon and summing the number of dots in each row. The first few pentagonal numbers are 1, 5, 12, 22, 35, and so on.

Pentagonal numbers have several interesting properties. They are also centered pentagonal numbers, meaning that they can be represented as a centered pentagon, a pentagon with a dot in the center and dots surrounding it in a pentagonal pattern. Additionally, pentagonal numbers are part of a broader class of figurate numbers, which include triangular, square, and hexagonal numbers, among others.

In number theory, pentagonal numbers play a role in various identities and relationships. For example, Euler's pentagonal number theorem states that every positive integer can be represented as the sum of at most two pentagonal numbers. This theorem has applications in combinatorics and number theory.

Pentagonal numbers also appear in various areas of mathematics and have been studied for their unique properties and relationships with other mathematical concepts.