View source for Hexagonal number

{{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".

==Test for hexagonal numbers==
One can efficiently test whether a positive integer ''x'' is a hexagonal number by computing

:<math>n = \frac{\sqrt{8x+1}+1}{4}.</math>

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

==Congruence relations==
* <math>h_n \equiv n \pmod{4}</math>
* <math>h_{3n}+h_{2n}+h_{n} \equiv 0 \pmod{2}</math>

==Other properties==
===Expression using sigma notation===
The ''n''th number of the hexagonal sequence can also be expressed by using [[sigma notation]] as

:<math> h_n = \sum_{k=0}^{n-1}{(4k+1)} </math>

where the [[empty sum]] is taken to be 0.

===Sum of the reciprocal hexagonal numbers===
The sum of the reciprocal hexagonal numbers is {{math|2ln(2)}}, where {{math|ln}} denotes [[natural logarithm]].
:<math>\begin{align} \sum_{k=1}^{\infty} \frac{1}{k(2k-1)} &= \lim_{n \to \infty}2\sum_{k=1}^{n} \left(\frac{1}{2k-1} - \frac{1}{2k} \right)\\                            &= \lim_{n \to \infty}2\sum_{k=1}^{n} \left(\frac{1}{2k-1} +  \frac{1}{2k} - \frac{1}{k} \right)\\ &= 2 \lim_{n \to \infty}\left(\sum_{k=1}^{2n}\frac{1}{k} - \sum_{k=1}^{n}\frac{1}{k}\right)\\                                                         &= 2 \lim_{n \to \infty}\sum_{k=1}^{n}\frac{1}{n+k} \\ &= 2 \lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\frac{k}{n}}\\ &= 2 \int_{0}^{1}\frac{1}{1+x}dx \\ &= 2 [ \ln(1+x) ]_{0}^{1} \\ &= 2 \ln{2}\\
                     & \approx{1.386294}
\end{align}</math>

===Multiplying the index===
Using rearrangement, the next set of formulas is given:
 
<math> h_{2n} = 4h_n+2n </math>

<math> h_{3n} = 9h_n+6n </math>

<math>...</math>

<math> h_{m*n} = m^{2}h_n+(m^{2}-m)n</math>

===Ratio relation===
Using the final formula from before with respect to ''m'' and then ''n'', and then some reducing and moving, one can get to the following equation:

<math>\frac{h_{m}+m}{h_{n}+n}=\left(\frac{m}{n}\right)^2</math>

===Numbers of divisors of powers of certain natural numbers===
<math>12^{n-1}</math> for ''n''>0 has <math>h_n</math> divisors.

Likewise, for any natural number of the form <math>r = p^2 q</math> where ''p'' and ''q'' are distinct prime numbers, <math> r^{n-1}</math> for ''n''>0 has <math>h_n</math> divisors.

''Proof.'' <math>r^{n-1} = (p^2 q)^{n-1} = p^{2(n-1)} q^{n-1} </math>
has divisors of the form <math>p^k q^l</math>, for ''k'' = 0 ... 2(''n'' &minus; 1), ''l'' = 0 ... n &minus; 1. Each combination of ''k'' and ''l'' yields a distinct divisor, so <math>r^{n-1}</math> has <math>[2(n - 1) + 1] [(n - 1) + 1]</math> divisors, i.e. <math> (2n - 1) n = h_n</math> divisors. ∎

== Hexagonal square numbers==

The sequence of numbers that are both hexagonal and perfect squares starts 1, 1225, 1413721,... {{OEIS2C|A046177}}.

==See also==
*[[Centered hexagonal number]]

==External links==
* {{MathWorld|urlname=HexagonalNumber|title=Hexagonal Number}}

{{Figurate numbers}}
{{Classes of natural numbers}}
{{series (mathematics)}}

[[Category:Figurate numbers]]