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Centered cube number
Centered figurate number that counts points in a three-dimensional pattern
Centered figurate number that counts points in a three-dimensional pattern
| Field | Value |
|---|---|
| image | Body centered cubic 35 balls.svg |
| image_size | 200px |
| caption | 35 points in a body-centered cubic lattice, forming two cubical layers around a central point. |
| number | Infinity |
| parentsequence | Polyhedral numbers |
| formula | n^3 + (n + 1)^3 |
| first_terms | 1, 9, 35, 91, 189, 341, 559 |
| OEIS | A005898 |
| OEIS_name | Centered cube |
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 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, ... .
Formulas
The centered cube number for a pattern with n concentric layers around the central point is given by the formula
:n^3 + (n + 1)^3 = (2n+1)\left(n^2+n+1\right).
The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as :\binom{(n+1)^2+1}{2}-\binom{n^2+1}{2} = (n^2+1)+(n^2+2)+\cdots+(n+1)^2.
Properties
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 numbers which are also the square numbers are 1 and 9,{{citation
References
References
- (2012). "Figurate Numbers". World Scientific.
- Lanski, Charles. (2005). "Concepts in Abstract Algebra". American Mathematical Society.
- {{Cite OEIS. A005898
- (2007). "The Magic Numbers of the Professor". Mathematical Association of America.
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