The centered octahedral numbers count the cubes used by this construction.

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".

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.

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.