Pentagonal Number Calculator

About this calculator

A pentagonal number is a figurate number that counts dots arranged in the shape of a nested pentagon, generated by the formula P(n) = n(3n − 1) ÷ 2. The sequence begins 1, 5, 12, 22, 35, 51... and, like triangular and square numbers, each term can be visualized as building outward layer by layer from a central point.

Pentagonal numbers are a standard topic in recreational and number theory, most famously appearing in Euler's pentagonal number theorem, which links them to the generating function for integer partitions. Students and math enthusiasts use a calculator like this to explore the sequence quickly, while the underlying identity has real applications in combinatorics and the study of partition functions.