Editing Centered hexagonal number (section)

{{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.