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centered octahedral number
A centered octahedral number or Haüy octahedral number is a figurate number that counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin.〔 The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths.〔 The Haüy octahedral numbers are named after René Just Haüy.
==History==
The name "Haüy octahedral number" comes from the work of René Just Haüy, a French mineralogist active in the late 18th and early 19th centuries. His "Haüy construction" approximates an octahedron as a polycube, formed by accreting concentric layers of cubes onto a central cube. The centered octahedral numbers count the number of cubes used by this construction. Haüy proposed this construction, and several related constructions of other polyhedra, as a model for the structure of crystalline minerals.〔. See in particular (p. 10 ).〕〔. See in particular (pp. 13–14 ). As cited by 〕
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