Centered octahedral number

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.

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]