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).
