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 i 2 points on the square faces of the i-th 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
Formulas
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]
Properties
Because of the factorization , it is impossible for a centered cube number to be a prime number.[3] The only centered cube number that is also a square number is 9.[4][5]
See also
References
- ↑ 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 .