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 areFormulas
The centered cube number for a pattern with concentric layers around the central point is given by the formula
The same number can also be expressed as a trapezoidal number, or a sum of consecutive numbers, asProperties
Because of the factorization, it is impossible for a centered cube number to be a prime number.
The only centered cube number that is also a square number is 9, which can be shown by solving.
