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 i2 points on the square faces of the ith 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 The centered cube number for a pattern with n concentric layers around the central point is given by the formula[1] The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as[2] Because of the factorization (2n + 1)(n2 + n + 1), it is impossible for a centered cube number to be a prime number.

[3] The only centered cube numbers which are also the square numbers are 1 and 9,[4][5] which can be shown by solving 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.