Named after | René Just Haüy |
---|---|
Publication year | 1801 |
Total no. of terms | Infinity |
Subsequence of |
Polyhedral numbers, Delannoy numbers |
Formula | |
First terms | 1, 7, 25, 63, 129, 231, 377 |
OEIS index |
|
A centered octahedral number or Haüy octahedral number is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. [1] The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. [2] The Haüy octahedral numbers are named after René Just Haüy.
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 cubes used by this construction. [3] Haüy proposed this construction, and several related constructions of other polyhedra, as a model for the structure of crystalline minerals. [4] [5]
The number of three-dimensional lattice points within n steps of the origin is given by the formula
The first few of these numbers (for n = 0, 1, 2, ...) are
The generating function of the centered octahedral numbers is [6] [7]
The centered octahedral numbers obey the recurrence relation [1]
They may also be computed as the sums of pairs of consecutive octahedral numbers.
The octahedron in the three-dimensional integer lattice, whose number of lattice points is counted by the centered octahedral number, is a metric ball for three-dimensional taxicab geometry, a geometry in which distance is measured by the sum of the coordinatewise distances rather than by Euclidean distance. For this reason, Luther & Mertens (2011) call the centered octahedral numbers "the volume of the crystal ball". [7]
The same numbers can be viewed as figurate numbers in a different way, as the centered figurate numbers generated by a pentagonal pyramid. That is, if one forms a sequence of concentric shells in three dimensions, where the first shell consists of a single point, the second shell consists of the six vertices of a pentagonal pyramid, and each successive shell forms a larger pentagonal pyramid with a triangular number of points on each triangular face and a pentagonal number of points on the pentagonal face, then the total number of points in this configuration is a centered octahedral number. [1]
The centered octahedral numbers are also the Delannoy numbers of the form D(3,n). As for Delannoy numbers more generally, these numbers count the paths from the southwest corner of a 3 × n grid to the northeast corner, using steps that go one unit east, north, or northeast. [2]