cp's OEIS Frontend

For all n > 0, the points at the 4 90-degree rotations of [n, 0, 0] and the eight 90-degree rotations and vertical reflections of [n/2, n/2, n*sqrt(1/2)] form the 12 vertices of a cuboctahedron (shown in red in the Links example). Points that otherwise lie on an axis plane have 24-point symmetry (green), and all other points have 48-point symmetry (blue). Thus, a(n) for all n > 0 are odd multiples of 12.

The number of noncongruent points for each sphere radius n (points within or on vertices or edges of each symmetry region in the Links example) gives A387222, the number of those points that are primitive for radius n (darker colors) gives A387223 for odd sphere radii, and the total primitive count divided by 12 gives A278081 for odd sphere radii. Examples of nonprimitive points include [3, 0, 0] and [3/2, 3/2, 3*sqrt(1/2)] for a(3), which reduce to a(1) primitive points [1, 0, 0] and [1/2, 1/2, sqrt(1/2)] respectively.

Analog for the simple cubic lattice is A016725.