Named after Haüy construction of an octahedron by 129 cubes René Just Haüy 1801 Infinity Polyhedral numbers,Delannoy numbers ${\displaystyle {\frac {(2n+1)\left(2n^{2}+2n+3\right)}{3))}$ 1, 7, 25, 63, 129, 231, 377 .mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}A001845Centered octahedral

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.

## 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 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]

## Formula

The number of three-dimensional lattice points within n steps of the origin is given by the formula

${\displaystyle {\frac {(2n+1)\left(2n^{2}+2n+3\right)}{3))}$

The first few of these numbers (for n = 0, 1, 2, ...) are

1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ...[6]

The generating function of the centered octahedral numbers is[6][7]

${\displaystyle {\frac {(1+x)^{3)){(1-x)^{4))}.}$

The centered octahedral numbers obey the recurrence relation[1]

${\displaystyle C(n)=C(n-1)+4n^{2}+2.}$

They may also be computed as the sums of pairs of consecutive octahedral numbers.

## Alternative interpretations

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]

## References

1. ^ a b c Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, pp. 107–109, 132, ISBN 9789814355483.
2. ^ a b Sulanke, Robert A. (2003), "Objects counted by the central Delannoy numbers" (PDF), Journal of Integer Sequences, 6 (1), Article 03.1.5, Bibcode:2003JIntS...6...15S, MR 1971435, retrieved September 8, 2014.
3. ^ Fathauer, Robert W. (2013), "Iterative arrangements of polyhedra – Relationships to classical fractals and Haüy constructions", Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (PDF)
4. ^ Maitte, Bernard (2013), "The Construction of Group Theory in Crystallography", in Barbin, Evelyne; Pisano, Raffaele (eds.), The Dialectic Relation Between Physics and Mathematics in the XIXth Century, History of Mechanism and Machine Science, vol. 16, Springer, pp. 1–30, doi:10.1007/978-94-007-5380-8_1, ISBN 9789400753808. See in particular p. 10.
5. ^ Haüy, René-Just (1784), Essai d'une théorie sur la structure des crystaux (in French). See in particular pp. 13–14. As cited by
6. ^ a b
7. ^ a b Luther, Sebastian; Mertens, Stephan (2011), "Counting lattice animals in high dimensions", Journal of Statistical Mechanics: Theory and Experiment, 2011 (9): P09026, arXiv:1106.1078, Bibcode:2011JSMTE..09..026L, doi:10.1088/1742-5468/2011/09/P09026, S2CID 119308823