Centered cube number

A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with points on the square faces of the th layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has 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, ... .

Formulas

The centered cube number for a pattern with concentric layers around the central point is given by the formula

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

The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as
:$\binom-\binom = (n^2+1)+(n^2+2)+\cdots+(n+1)^2.$

Properties

Because of the factorization , it is impossible for a centered cube number to be a prime number.
The only centered cube numbers which are also the square numbers are 1 and 9,. which can be shown by solving , the only integer solutions being (x,y) from , 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 where a,b are half-integers.

See also

* Cube number

References

External links

*

{{Classes of natural numbers

Figurate numbers