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
- 1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... (sequence A005898 in the OEIS).
The centered cube number for a pattern with n concentric layers around the central point is given by the formula
Because of the factorization (2n + 1)(n2 + n + 1), 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 2n + 1 = n2 + n + 1.
- Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, pp. 121–123, ISBN 9789814355483
- Lanski, Charles (2005), Concepts in Abstract Algebra, American Mathematical Society, p. 22, ISBN 9780821874288.
- Sloane, N. J. A. (ed.). "Sequence A005898". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- Stroeker, R. J. (1995), "On the sum of consecutive cubes being a perfect square", Compositio Mathematica, 97 (1–2): 295–307, MR 1355130.
- O'Shea, Owen; Dudley, Underwood (2007), The Magic Numbers of the Professor, MAA Spectrum, Mathematical Association of America, p. 17, ISBN 9780883855577.