Centered octahedral number
{{Short description|Centered figurate number representing an octahedron}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
{{Infobox integer sequence
| image = Haüy construction 129.svg
| image_size = 200px
| alt =
| caption = Haüy construction of an octahedron by 129 cubes
| named_after = René Just Haüy
| publication_year = 1801
| number = Infinity
| parentsequence = Polyhedral numbers,
Delannoy numbers
| formula =
| first_terms = 1, 7, 25, 63, 129, 231, 377
| OEIS = A001845
| OEIS_name = Centered octahedral
}}
In mathematics, 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. 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 cubes used by this construction.{{citation |contribution=Iterative arrangements of polyhedra – Relationships to classical fractals and Haüy constructions |first=Robert W. |last=Fathauer |url=http://archive.bridgesmathart.org/2013/bridges2013-175.pdf |year=2013 |title=Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture }} Haüy proposed this construction, and several related constructions of other polyhedra, as a model for the structure of crystalline minerals.{{citation |title=The Dialectic Relation Between Physics and Mathematics in the XIXth Century |volume=16 |series=History of Mechanism and Machine Science |editor1-first=Evelyne |editor1-last=Barbin |editor2-first=Raffaele |editor2-last=Pisano |publisher=Springer |year=2013 |isbn=9789400753808 |contribution=The Construction of Group Theory in Crystallography |pages=1–30 |first=Bernard |last=Maitte |doi=10.1007/978-94-007-5380-8_1 }}. See in particular [https://books.google.com/books?id=qaREAAAAQBAJ&pg=PA10 p. 10].{{citation |last=Haüy|first=René-Just |title=Essai d'une théorie sur la structure des crystaux |language=French |year=1784 }}. See in particular [https://books.google.com/books?id=ORwUAAAAQAAJ&pg=PA13 pp. 13–14]. As cited by {{mathworld |id=HauyConstruction |title=Haűy [sic] Construction }}
Formula
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
:1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ...{{Cite OEIS|A001845 |name=Centered octahedral numbers (crystal ball sequence for cubic lattice) }}
The generating function of the centered octahedral numbers is{{citation
| last1 = Luther | first1 = Sebastian
| last2 = Mertens | first2 = Stephan
| arxiv = 1106.1078
| issue = 9
| journal = Journal of Statistical Mechanics: Theory and Experiment
| page = 09026
| title = Counting lattice animals in high dimensions
| url = http://stacks.iop.org/1742-5468/2011/i=09/a=P09026
| volume = 2011
| year = 2011 | doi = 10.1088/1742-5468/2011/09/P09026
| bibcode = 2011JSMTE..09..026L
| s2cid = 119308823
}}
:
The centered octahedral numbers obey the recurrence relation
:
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, {{harvtxt |Luther |Mertens |2011 }} call the centered octahedral numbers "the volume of the crystal ball".
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.{{citation |pages=107–109, 132 |title=Figurate Numbers |first1=Elena |last1=Deza|author1-link=Elena Deza |first2=Michel |last2=Deza |author2-link=Michel Deza |publisher=World Scientific |year=2012 |isbn=9789814355483 |url=https://books.google.com/books?id=cDxYdstLPz4C&pg=PA107 }}.
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.{{citation
| last = Sulanke
| first = Robert A.
| issue = 1
| journal = Journal of Integer Sequences
| mr = 1971435
| at = Article 03.1.5
| title = Objects counted by the central Delannoy numbers
| url = http://www.emis.de/journals/JIS/VOL6/Sulanke/delannoy.pdf
| volume = 6
| year = 2003
| bibcode = 2003JIntS...6...15S
| access-date = 2014-09-08
}}.