Centered cube number

{{Short description|Centered figurate number that counts points in a three-dimensional pattern}}

{{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 numbers

| formula = $n^3 + (n + 1)^3$

| first_terms = 1, 9, 35, 91, 189, 341, 559

| OEIS = A005898

| OEIS_name = Centered cube

}}

A centered cube number is a centered figurate number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with {{math|i²}} 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, 9, 35, 91, 189, 341, 559, 855, 1241, 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{{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}}

: $n^3 + (n + 1)^3 = (2n+1)\left(n^2+n+1\right).$

The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as{{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}}.

: $\binom{(n+1)^2+1}{2}-\binom{n^2+1}{2} = (n^2+1)+(n^2+2)+\cdots+(n+1)^2.$

Properties

Because of the factorization {{math|(2n + 1)(n² + n + 1)}}, it is impossible for a centered cube number to be a prime number.{{Cite OEIS|A005898}}

The only centered cube numbers which are also the square numbers are 1 and 9,{{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}}.{{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}}. which can be shown by solving {{math|x² {{=}} y³ + 3y }}, 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.

References

External links

{{mathworld|id=CenteredCubeNumber|title=Centered Cube Number}}

Category:Figurate numbers

Centered cube number

Formulas

Properties

See also

References

External links