A polygonal number of the form . The first few are 1, 5, 12, 22, 35, 51, 70, ... (OEIS A000326). The generating function for the pentagonal numbers is

Every pentagonal number is 1/3 of a triangular number.

The so-called generalized pentagonal numbers are given by with , , , ..., the first few of which are 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (OEIS A001318).

There are conjectured to be exactly 210 positive integers that cannot be represented using three pentagonal numbers, namely 4, 8, 9, 16, 19, 20, 21, 26, 30, 31, 33, 38, 42, 43, 50, 54, ..., 20250, 33066, (OEIS A007527; Guy 1994a).

There are six positive integers that cannot be expressed using four pentagonal numbers: 9, 21, 31, 43, 55, and 89 (OEIS A133929).

All positive integers can be expressed using five pentagonal numbers.

Letting be the set of numbers relatively prime to 6, the generalized pentagonal numbers are given by . Also, letting be the subset of the for which , the usual pentagonal numbers are given by (D. Terr, pers. comm., May 20, 2004).