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| A '''centered hexagonal number''', or '''hex number''', 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]].
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| ! 1 !! !! 7 !! !! 19 !! !! 37
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| | +1</span> || || +6 || || +12 || || +18
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| | Introduction. The author's name is Dalton but nonetheless , it's not the most masucline name out [http://Www.Tumblr.com/tagged/presently presently]. To drive is one of the things he loves the vast majority. His wife and him chose to live on in South Carolina yet his family loves that will. Auditing is where his primary income comes from. He can be running and maintaining one specific blog here: http://prometeu.net<br><br>my page; [http://prometeu.net clash of clans hack tool v3.1 password] |
| |[[Image:RedDotX.svg|16px|*]]
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| |[[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|*]]
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| |[[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|*]]
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| |[[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|*]]
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| The {{mvar|n}}th centered hexagonal number is given by the formula | |
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| :<math>n^3 - (n-1)^3 = 3n(n-1)+1.\,</math> | |
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| [[Image:Centered hexagonal = 1 + 6triangular.svg|thumb|right]]
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| Expressing the formula as
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| :<math>1+6\left({1\over 2}n(n-1)\right)</math>
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| shows that the centered hexagonal number for {{mvar|n}} is 1 more than 6 times the {{math|(''n'' − 1)}}th [[triangular number]].
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| The first few centered hexagonal numbers are
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| :[[1 (number)|1]], [[7 (number)|7]], [[19 (number)|19]], [[37 (number)|37]], [[61 (number)|61]], [[91 (number)|91]], [[127 (number)|127]], [[169 (number)|169]], 217, 271, 331, 397, 469, 547, 631, 721, 817, 919.
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| In [[base 10]] one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1.
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| Centered hexagonal numbers have practical applications in materials logistics management, for example, in [[packing problem|packing]] round items into larger round containers, such as [[Vienna sausage]]s into round cans, or combining individual [[wire]] strands into a [[cable]].
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| The sum of the first {{mvar|n}} centered hexagonal numbers happens to be [[cube (algebra)|{{math|''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. In particular, [[prime number|prime]] centered hexagonal numbers are [[cuban prime]]s.
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| 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]].
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| ==See also==
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| *[[Hexagonal number]]
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| *[[Flower of Life]] (generated by circles arranged in a centered hexagonal pattern)
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| *[[Magic hexagon]]
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| *[[Star number]]
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Centered Hexagonal Number}}
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| [[Category:Figurate numbers]]
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Introduction. The author's name is Dalton but nonetheless , it's not the most masucline name out presently. To drive is one of the things he loves the vast majority. His wife and him chose to live on in South Carolina yet his family loves that will. Auditing is where his primary income comes from. He can be running and maintaining one specific blog here: http://prometeu.net
my page; clash of clans hack tool v3.1 password