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 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
The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as
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.
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