Pentagonal numbers are figurate numbers based on a pentagon, given by \(P_k = k(3k - 1)/2\) for \(k \geq 1\). First terms: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.
Geometrically, \(P_k\) counts dots arranged in a pentagonal pattern of side \(k\) that shares corners with previous pentagons.
Euler's Pentagonal Number Theorem is a famous identity relating pentagonal numbers to the partition function — a foundational result in the theory of integer partitions and modular forms.