Hexagonal number

A hexagonal number is a figurate number. The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side.

The formula for the nth hexagonal number

$h_n= 2n^2-n = n(2n-1) = {{2n}\times{(2n-1)}\over 2}.\,\!$

The first few hexagonal numbers (sequence A000384 in 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 2x3 = 6; the 4th is 4x7 = 28; the 16th is 16x31 = 496; and the 64th is 64x127 = 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 (2n−1).

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

Test for hexagonal numbers

One can efficiently test whether a positive integer x is an hexagonal number by computing

$n = \frac{\sqrt{8x+1}+1}{4}.$

If n is an integer, then x is the nth hexagonal number. If n is not an integer, then x is not hexagonal.

Other properties

The nth number of the hexagonal sequence can also be expressed by using Sigma notation as

$h_n = \sum_{i=0}^{n-1}{(4i+1)}$

where the empty sum is taken to be 0.

See also

• Centered hexagonal number

External links

• Mathworld entry on Hexagonal Number