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 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
Formulas
The centered cube number for a pattern with concentric layers around the central point is given by the formulaThe same number can also be expressed as a trapezoidal number, or a sum of consecutive numbers, as
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 from, By substituting a=/2 and b=y/2, we obtain x^2=2y^3+3y^2+3y+1. This gives only from where a,b are half-integers.