Talk:Centered octahedral number

Crystallography
In crystallography the habit of a crystal belonging to a cubic space group may be either cubic or octahedral as both both cube and octahedron belong to the same point group, Oh. Another way to obtain these numbers would be to inscribe an octahedron in a cube (vertices at the mid-points of the sides of the cube), then shave off the 8 corners of the cube until the octahedron is obtained. Petergans (talk) 16:07, 22 September 2014 (UTC)

Orthoplexes in higher dimensions
I am considering adding a section to cover the general case of an Orthoplex construction in arbitrary dimension, which can go down to 0 (where the value is always 1), 1 Dimension (all odd numbers), 2 Dimensions (1, 5, 13, 25, 41, etc.) and the higher dimensional constructions which align with the crystal ball constructions and other permutations of the (x + 1)dimension/(x - 1)(dimension + 1) expansion, and the general extension of the cuboid constructions to these higher dimensions in keeping with various (incomplete) OEIS sequences. For example, in 4 dimensions, this sequence is 1, 9, 41, 129, 321, 681, etc.

There is also a diagonal symmetry to the sequence when plotted in a grid of Dimension by Number of Elements such that the lines where the dimension and the number sum to a constant, that diagonal will be mirrored, its greatest value at where dimension and number are equal. This is a basic property of the Delannoy numbers, wherein the above values derive.

Would this be of interest to the general readership?

TimeHorse (talk) 20:36, 24 March 2022 (UTC)
 * Only with published sources (not just OEIS sequences) attesting to its significance with respect to the main topic of this article, the centered octahedral numbers. Do you have such sources? ——David Eppstein (talk) 21:26, 24 March 2022 (UTC)