A pentagonal number is a polygonal number of the form 
, where 
is a positive integer.
The first few are 1, 5, 12, 22, 35, 51, 70, ... (OEIS A000326).
The generating function for the pentagonal
numbers is
 |
for 
.
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). In increasing
order, these numbers are
 |
for a nonnegative integer 
. In particular, 
for 
and 
for 
.
There are conjectured to be exactly 210 positive integers that cannot be represented as a sum of at most three pentagonal numbers, beginning 4, 8, 9, 16, 19, 20, 21, 26, 30, 31, 33, 38, 42, 43, 50, 54, ... and ending with 20250 and 33066 (OEIS A003679; Guy 1994a).
There are six positive integers that cannot be expressed as a sum of at most four pentagonal numbers: 9, 21, 31, 43, 55, and 89 (OEIS A133929).
All positive integers can be expressed as a sum of at most five pentagonal numbers.
Letting 
run through the positive integers relatively
prime to 6, the generalized pentagonal numbers are given by 
. Also, letting 
be those 
for which 
, the usual pentagonal numbers are given by 
(D. Terr, pers. comm.,
May 20, 2004).
