A hexagonal number is a polygonal number and 6-polygonal number of
the form 
for a positive integer 
. The first few are 1, 6, 15, 28, 45, ... (OEIS A000384).
The generating function for the hexagonal
numbers is given by
 |
(1)
|
for 
.
The related sequence 
for 
, 1, 2, ... gives 0, 3, 10, 21, 36, ... (OEIS A014105),
and is obtained by replacing 
by 
in the hexagonal number formula. It also satisfies
 |
(2)
|
where 
is the floor function. The fractional
part of the sum tends to 
as 
(Awasthi and Mishra 2026).
Every hexagonal number is a triangular number since
![r(2r-1)=1/2(2r-1)[(2r-1)+1].](/images/equations/HexagonalNumber/NumberedEquation3.svg) |
(3)
|
Duke (1997, p. 191) describes Legendre's 1830 proof that every integer larger than 1791 is a sum of at most four hexagonal numbers. Duke and Schulze-Pillot (1990) improved this to three hexagonal numbers for every sufficiently large integer.
There are exactly 13 positive integers that cannot be represented using at most four hexagonal numbers, namely 5, 10, 11, 20, 25, 26, 38, 39, 54, 65, 70, 114, and 130 (OEIS A007527; Guy 1994a).
Similarly, there are only two positive integers that cannot be represented using at most five hexagonal numbers, namely
 |  |  |
(4)
|
 |  |  |
(5)
|
Every positive integer can be represented using at most six hexagonal numbers.
th Roots of Integers Partitioned by the 
th Powers." Math. Gaz. 110, 116-121, 2026.