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 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:#d20; vertical-align:middle;" | +1 || || +6 || || +12 || || +18

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Centered hexagonal numbers should not be confused with 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}}:

{{bi|1=1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 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 parallelograms of {{math|''n''(''n''−1)}} dots each.]]

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

{{bi|1=<math>H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \,</math>}}

Expressing the formula as

{{bi|1=<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

{{bi|1=<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/>

{{bi|1=<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

{{bi|1=<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

{{bi|1=<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

{{bi|1=<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''³ 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 period 5). This follows from the 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: 1₆, 11₆, 31₆, 101₆, 141₆, 231₆, 331₆, 441₆... This follows from the fact that every centered hexagonal number modulo 6 (=10₆) equals 1.

The sum of the first {{mvar|n}} centered hexagonal numbers is {{math|''n''³}}. That is, centered hexagonal pyramidal numbers and 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 gnomon of the cubes. (This can be seen geometrically from the diagram.) In particular, prime centered hexagonal numbers are cuban primes.

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

== Applications == === Telescopes === [[File:comparison_optical_telescope_primary_mirrors.svg|thumb|Ignoring central holes, the number of mirror segments in several segmented mirror telescopes are centered hexagonal numbers]] Many segmented mirror reflecting telescopes 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: {{table alignment}} {| class="wikitable defaultcenter col1left" ! Telescope !! Number of<br />segments !! Number<br />missing !! Total !! ''n''-th centered<br />hexagonal number |- | Giant Magellan Telescope || &numsp;7 || 0 || '''&numsp;7''' || 2 |- | James Webb Space Telescope || 18 || 1 || '''19''' || 3 |- | Gran Telescopio Canarias || 36 || 1 || '''37''' || 4 |- | Guido Horn d'Arturo's prototype || 61 || 0 || '''61''' || 5 |- | Southern African Large Telescope || 91 || 0 || '''91''' || 6 |} === Rocket pods === [[File:Hydra 70 03.jpg|thumb|right|The M261 rocket pod has a capacity of 19 Hydra 70 rockets, arranged hexagonally.]] Many rocket pods in service currently use centered hexagonal numbers of rocket tubes. {| class="wikitable defaultcenter col1left" |- ! Rocket Pod !! Capacity !! ''n''-th centered hexagonal number |- | M260 || &numsp;7 || 2 |- | M261 || 19 || 3 |}

==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