A centered octahedral number or Haüy octahedral number is a

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...

that counts the number of points of a three-dimensional

integer lattice In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...

that lie inside an

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

centered at the origin. The same numbers are special cases of the

Delannoy number In mathematics, a Delannoy number D describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (''m'', ''n''), using only single steps north, northeast, or east. The Delannoy numbers are named aft ...

s, which count certain two-dimensional lattice paths. The Haüy octahedral numbers are named after

René Just Haüy René Just Haüy () FRS MWS FRSE (28 February 1743 – 1 June 1822) was a French priest and mineralogist, commonly styled the Abbé Haüy after he was made an honorary canon of Notre Dame. Due to his innovative work on crystal structure and hi ...

History

The name "Haüy octahedral number" comes from the work of

, a French

mineralogist Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical) properties of minerals and mineralized artifacts. Specific studies within mineralogy include the proce ...

active in the late 18th and early 19th centuries. His "Haüy construction" approximates an octahedron as a

polycube upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total A puzzle involving arranging nine L tricubes into a 3×3 cube A polycube is a solid figure formed by j ...

, 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 particula
pp. 13–14
As cited by

Formula

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

\frac

The first few of these numbers (for ''n'' = 0, 1, 2, ...) are :1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, ... The generating function of the centered octahedral numbers is :

\frac.

The centered octahedral numbers obey the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

C(n)=C(n-1)+4n^2+2.

They may also be computed as the sums of pairs of consecutive

octahedral number In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahed ...

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 taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian c ...

, a geometry in which distance is measured by the sum of the coordinatewise distances rather than by

Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...

. For this reason, 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 In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. The ''regular'' pentagonal pyramid has a base that is a regu ...

. 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 A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...

of points on each triangular face and a

pentagonal number A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...

of points on the pentagonal face, then the total number of points in this configuration is a centered octahedral number.. The centered octahedral numbers are also the

s of the form ''D''(3,''n''). As for Delannoy numbers more generally, these numbers count the number of paths from the southwest corner of a 3 × ''n'' grid to the northeast corner, using steps that go one unit east, north, or northeast..

References

{{Classes of natural numbers Figurate numbers