Editing Centered cube number

{{Short description|Centered figurate number that counts points in a three-dimensional pattern}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
{{Infobox integer sequence
| image = Body centered cubic 35 balls.svg
| image_size = 200px
| alt = 
| caption = 35 points in a body-centered cubic lattice, forming two cubical layers around a central point.
| number = [[Infinity]]
| parentsequence = [[Polyhedral number]]s
| formula = <math>n^3 + (n + 1)^3</math>
| first_terms = [[1]], [[9]], [[35 (number)|35]], [[91 (number)|91]], [[189 (number)|189]], [[341 (number)|341]], [[559 (number)|559]]
| OEIS = A005898
| OEIS_name = Centered cube
}}
A '''centered cube number''' is a [[centered number|centered]] [[figurate number]] that counts the points in a three-dimensional pattern formed by a point surrounded by concentric [[cube|cubical]] layers of points, with {{math|''i''<sup>2</sup>}} points on the square faces of the {{mvar|i}}th layer. Equivalently, it is the number of points in a [[body-centered cubic]] pattern within a cube that has {{math|''n'' + 1}} points along each of its edges.

The first few centered cube numbers are

:[[1 (number)|1]], [[9 (number)|9]], [[35 (number)|35]], [[91 (number)|91]], [[189 (number)|189]], 341, 559, 855, 1241, [[1729 (number)|1729]], 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... {{OEIS|id=A005898}}.

==Formulas==
The centered cube number for a pattern with {{mvar|n}} concentric layers around the central point is given by the formula<ref>{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|author1-link=Elena Deza|first2=Michel|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|pages=121–123|url=https://books.google.com/books?id=cDxYdstLPz4C&pg=PA121}}</ref>

:<math>n^3 + (n + 1)^3 = (2n+1)\left(n^2+n+1\right).</math>

The same number can also be expressed as a [[trapezoidal number]] (difference of two [[triangular number]]s), or a sum of consecutive numbers, as<ref>{{citation|title=Concepts in Abstract Algebra|first=Charles|last=Lanski|publisher=American Mathematical Society|year=2005|isbn=9780821874288|page=22|url=https://books.google.com/books?id=X1ttNRvbNK0C&pg=PA22}}.</ref>
:<math>\binom{(n+1)^2+1}{2}-\binom{n^2+1}{2} = (n^2+1)+(n^2+2)+\cdots+(n+1)^2.</math>

==Properties==
Because of the factorization {{math|(2''n'' + 1)(''n''<sup>2</sup> + ''n'' + 1)}}, it is impossible for a centered cube number to be a [[prime number]].<ref>{{Cite OEIS|A005898}}</ref>
The only centered cube numbers which are also the [[square number]]s are&nbsp;1 and 9,<ref>{{citation
 | last = Stroeker | first = R. J.
 | title = On the sum of consecutive cubes being a perfect square
 | journal = [[Compositio Mathematica]]
 | volume = 97
 | year = 1995
 | issue = 1–2
 | pages = 295–307
 | mr = 1355130
 | url = http://www.numdam.org/item?id=CM_1995__97_1-2_295_0}}.</ref><ref>{{citation|title=The Magic Numbers of the Professor|series=MAA Spectrum|first1=Owen|last1=O'Shea|first2=Underwood|last2=Dudley|publisher=Mathematical Association of America|year=2007|isbn=9780883855577|page=17|url=https://books.google.com/books?id=RC9304k036YC&pg=PA17}}.</ref> which can be shown by solving {{math|''x''<sup>2</sup> {{=}} ''y''<sup>3</sup> + 3''y'' }}, the only integer solutions being (x,y) from {(0,0), (1,2), (3,6), (12,42)}, By substituting a=(x-1)/2 and b=y/2, we obtain x^2=2y^3+3y^2+3y+1. This gives only (a,b) from {(-1/2,0), (0,1), (1,3), (11/2,21)} where a,b are half-integers.

==See also==
* [[Cube number]]

==References==
{{reflist}}

==External links==
*{{mathworld|id=CenteredCubeNumber|title=Centered Cube Number}}

{{Figurate numbers}}
{{Classes of natural numbers}}

[[Category:Figurate numbers]]