Editing Hexagonal number (section)

{{Short description|Type of figurate number}}
{{distinguish|text=[[Hexadecimal]] numbers}}
[[File:hexagonal_number_visual_proof.svg|thumb|[[Proof without words]] that a hexagonal number (middle column) can be rearranged as rectangular and odd-sided triangular numbers]]
A '''hexagonal number''' is a [[figurate number]]. The ''n''th hexagonal number ''h''<sub>''n''</sub> 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 (geometry)|vertex]].
[[Image:Hexagonal_numbers.svg|none|The first four hexagonal numbers.]]
The formula for the ''n''th hexagonal number

:<math>h_n= 2n^2-n = n(2n-1) = \frac{2n(2n-1)}{2}.</math>

The first few hexagonal numbers {{OEIS|id=A000384}} are:

:[[1 (number)|1]], [[6 (number)|6]], [[15 (number)|15]], [[28 (number)|28]], [[45 (number)|45]], [[66 (number)|66]], [[91 (number)|91]], [[120 (number)|120]], [[153 (number)|153]], [[190 (number)|190]], [[231 (number)|231]], [[276 (number)|276]], [[325 (number)|325]], 378, 435, [[496 (number)|496]], [[561 (number)|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 
:<math>M_p 2^{p-1} = M_p \frac{M_p + 1}{2} = h_{(M_p+1)/2}=h_{2^{p-1}}</math>
:where ''M''<sub>''p''</sub> 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 (number)|130]]. [[Adrien-Marie Legendre]] proved in 1830 that any [[integer]] greater than 1791 can be expressed in this way.

In addition, only two integers cannot be expressed using five hexagonal numbers (but can be with six), those being 11 and 26.

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