Centered cube number

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 $i^{2}$ points on the square faces of the $i$th layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has $n + 1$ 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, ....

Formulas
The centered cube number for a pattern with $n$ 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
 * $$\binom{(n+1)^2+1}{2}-\binom{n^2+1}{2} = (n^2+1)+(n^2+2)+\cdots+(n+1)^2.$$

Properties
Because of the factorization $(2n + 1)(n^{2} + n + 1)$, 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 $x^{2} = y^{3} + 3y$, 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.