Editing Centered hexagonal number

{{Short description|Number that represents a hexagon with a dot in the center}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
[[File:Catan_Universe_fixed_setup.svg|thumb|Centered hexagonal numbers appearing in the [[Catan]] board game:<br />19 land tiles,<br />37 total tiles]]
In [[mathematics]] and [[combinatorics]], a '''centered hexagonal number''', or '''centered''' '''hexagon number''',<ref>{{cite journal|last = Hindin|first = H. J.|title = Stars, hexes, triangular numbers and Pythagorean triples|journal = J. Rec. Math.|volume = 16| pages = 191–193|date=1983}}</ref><ref name=Deza>{{Cite book|last1=Deza|first1=Elena|author1-link=Elena Deza|url=https://books.google.com/books?id=cDxYdstLPz4C|title=Figurate Numbers|last2=Deza|first2=M.|date=2012|publisher=World Scientific|isbn=978-981-4355-48-3|language=en|pages=47–55}}</ref> is a [[centered polygonal number|centered]] [[figurate number]] that represents a [[hexagon]] with a dot in the center and all other dots surrounding the center dot in a [[hexagonal lattice]]. The following figures illustrate this arrangement for the first four centered hexagonal numbers:
:{|style="min-width: 325px;"|
! 1 !! !! 7 !! !! 19 !! !! 37
|-  style="text-align:center; color:red; vertical-align:middle;"
| +1 || || +6 || || +12 || || +18

|-  style="vertical-align:middle; text-align:center; line-height:1.1em;"
|[[Image:RedDotX.svg|16px|*]]
|
|[[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]
|
|[[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]
|
|[[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:GrayDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]] [[Image:RedDotX.svg|16px|*]]
|}

Centered hexagonal numbers should not be confused with [[hexagonal number|cornered hexagonal numbers]], which are figurate numbers in which the associated hexagons share a vertex.

The sequence of hexagonal numbers starts out as follows {{OEIS|id=A003215}}:

:[[1]], [[7]], [[19 (number)|19]], [[37 (number)|37]], [[61 (number)|61]], [[91 (number)|91]], [[127 (number)|127]], [[169 (number)|169]], [[217 (number)|217]], [[271 (number)|271]], [[331 (number)|331]], [[397 (number)|397]], 469, 547, 631, 721, 817, 919.

== Formula ==
[[Image:Centered hexagonal = 1 + 6triangular.svg|thumb|right|Dissection of hexagonal number into six triangles with a remainder of one.  The triangles can be re-assembled pairwise to give three [[parallelogram]]s of {{math|''n''(''n''−1)}} dots each.]]

The {{mvar|n}}th centered hexagonal number is given by the formula<ref name=Deza/>

:<math>H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \,</math>

Expressing the formula as

:<math>H(n) = 1+6\left(\frac{n(n-1)}{2}\right)</math>

shows that the centered hexagonal number for {{mvar|n}} is 1 more than 6 times the {{math|(''n'' − 1)}}th [[triangular number]].

In the opposite direction, the ''index'' {{mvar|n}} corresponding to the centered hexagonal number <math>H = H(n)</math> can be calculated using the formula

:<math>n=\frac{3+\sqrt{12H-3}}{6}.</math>

This can be used as a test for whether a number {{mvar|H}} is centered hexagonal: it will be if and only if the above expression is an integer.

== Recurrence and generating function ==
The centered hexagonal numbers <math>H(n)</math> satisfy the [[recurrence relation]]<ref name=Deza/>

:<math>H(n+1) = H(n) + 6n.</math>

From this we can calculate the [[generating function]] <math>F(x) = \sum_{n \ge 0} H(n) x^n</math>. The generating function satisfies

:<math>F(x) = x + xF(x) + \sum_{n \ge 2} 6n x^n.</math>

The latter term is the [[Taylor series]] of <math>\frac{6x}{(1-x)^2} - 6x</math>, so we get

:<math>(1 - x) F(x) = x + \frac{6x}{(1-x)^2} - 6x = \frac{x + 4x^2 + x^3}{(1-x)^2}</math>

and end up at

:<math>F(x) = \frac{x + 4x^2 + x^3}{(1-x)^3}.</math>

== Properties ==
[[File:visual_proof_centered_hexagonal_numbers_sum.svg|thumb|[[Proof without words]] of the sum of the first ''n'' hex numbers by arranging ''n''<sup>3</sup> semitransparent balls in a cube and viewing along a [[space diagonal]] &ndash; colour denotes cube layer and line style denotes hex number]]
In [[base 10]] one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with [[Periodic sequence|period]] 5).
This follows from the [[Triangular number#Other properties|last digit of the triangle numbers]] {{OEIS|id=A008954}} which repeat 0-1-3-1-0 when taken modulo 5.
In [[base 6]] the rightmost digit is always 1: 1<sub>6</sub>, 11<sub>6</sub>, 31<sub>6</sub>, 101<sub>6</sub>, 141<sub>6</sub>, 231<sub>6</sub>, 331<sub>6</sub>, 441<sub>6</sub>...
This follows from the fact that every centered hexagonal number modulo 6 (=10<sub>6</sub>) equals 1.

The sum of the first {{mvar|n}} centered hexagonal numbers is {{math|[[cube (algebra)|''n''<sup>3</sup>]]}}. That is, centered hexagonal [[pyramidal number]]s and [[cubic number|cubes]] are the same numbers, but they represent different shapes.  Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the [[figurate number#Gnomon|gnomon]] of the cubes. (This can be seen geometrically from the diagram.)  In particular, [[prime number|prime]] centered hexagonal numbers are [[cuban prime]]s.

The difference between {{math|(2''n'')<sup>2</sup>}} and the {{mvar|n}}th centered hexagonal number is a number of the form {{math|3''n''<sup>2</sup> + 3''n'' − 1}}, while the difference between {{math|(2''n'' − 1)<sup>2</sup>}} and the {{mvar|n}}th centered hexagonal number is a [[pronic number]].

== Applications ==
[[File:comparison_optical_telescope_primary_mirrors.svg|thumb|Ignoring central holes, the number of mirror segments in several [[segmented mirror]] [[telescope]]s are centered hexagonal numbers]]
Many [[segmented mirror]] [[reflecting telescope]]s have primary mirrors comprising a centered hexagonal number of segments (neglecting the central segment removed to allow passage of light) to simplify the control system.<ref>Mast, T. S. and Nelson, J. E. [http://osti.gov/servlets/purl/6194407 ''Figure control for a segmented telescope mirror'']. United States: N. p., 1979. Web. doi:10.2172/6194407.</ref> Some examples:
{| class="wikitable" style="text-align:center;"
! Telescope !! Number of<br />segments !! Number<br />missing !! Total !! ''n''-th centered<br />hexagonal number
|-
| align="left"|[[Giant Magellan Telescope]]         ||  7 || 0 ||  '''7''' || 2
|-
| align="left"|[[James Webb Space Telescope]]       || 18 || 1 || '''19''' || 3
|-
| align="left"|[[Gran Telescopio Canarias]]         || 36 || 1 || '''37''' || 4
|-
| align="left"|[[Guido Horn d'Arturo]]'s prototype  || 61 || 0 || '''61''' || 5
|-
| align="left"|[[Southern African Large Telescope]] || 91 || 0 || '''91''' || 6
|}

==References==
{{reflist}}

==See also==
*[[Hexagonal number]]
*[[Magic hexagon]]
*[[Star number]]

{{Figurate numbers}}
{{Classes of natural numbers}}
{{Authority control}}

{{DEFAULTSORT:Centered Hexagonal Number}}
[[Category:Figurate numbers]]
[[Category:Integer sequences]]