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.

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.

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

(
2
n
+
1
)
×
(
2
n
2
+
2
n
+
3
)
3
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

(
1
+
x
)
3
(
1
−
x
)
4
.
The centered octahedral numbers obey the recurrence relation

C
(
n
)
=
C
(
n
−
1
)
+
4
n
2
+
2.

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".

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.

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 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.