A **hexagonal number** is a figurate number. The *n*th hexagonal number *h*_{n} is the number of *distinct* dots in a pattern of dots consisting of the *outlines* of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex.

The formula for the *n*th hexagonal number

h
n
=
2
n
2
−
n
=
n
(
2
n
−
1
)
=
2
n
×
(
2
n
−
1
)
2
.
The first few hexagonal numbers (sequence A000384 in the OEIS) are:

1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946.

Every hexagonal number is a triangular number, but only every *other* triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9".

Every even perfect number is hexagonal, given by the formula

M
p
2
p
−
1
=
M
p
(
M
p
+
1
)
/
2
=
h
(
M
p
+
1
)
/
2
=
h
2
p
−
1
where

*M*_{p} is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are hexagonal.
For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128.

The largest number that cannot be written as a sum of at most four hexagonal numbers is 130. Adrien-Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way.

Hexagonal numbers can be rearranged into rectangular numbers of size *n* by (2*n*−1).

Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".