A387223 Number of noncongruent primitive points in a face-centered cubic lattice that intersect a sphere of radius 2*n-1 centered on a point in the lattice.

1, 1, 2, 2, 2, 3, 4, 3, 5, 5, 4, 6, 8, 5, 8, 8, 6, 12, 10, 7, 11, 11, 9, 12, 14, 9, 14, 15, 10, 15, 16, 12, 22, 17, 12, 18, 19, 15, 20, 20, 14, 21, 25, 15, 23, 28, 16, 27, 25, 16, 26, 26, 24, 27, 28, 19, 29, 36, 21, 32, 28, 21, 38, 32, 22, 33, 36, 27, 35, 35

Offset

1,3

Comments

Here a primitive point is one which is not an integer multiple of a point that intersects a sphere of a smaller integer radius, e.g. [13.5, 1.5, 9*sqrt(1/2)] is not primitive for sphere radius 15 (n = 8) because it is a multiple of [4.5, 0.5, 3*sqrt(1/2)] which is primitive for sphere radius 5 (n = 3).

Subset of A387222, excluding nonprimitive points, and omitting spheres with even radii (radius 0 has only the 1 center point, and all others have no primitive points).

It is observed that for most n >= 6, at least 1 combination of at least 3 of the a(n) constituent points, as n*2-1 length vectors, can be arranged via cuboctahedral rotations and reflections to form a closed walk. E.g. for n = 6, the 3 noncongruent vectors of length 6*2-1 = 11 can be arranged as [5/2, 19/2, 7*sqrt(1/2)] + [8, -7, -4*sqrt(1/2)] + [-21/2, -5/2, -3*sqrt(1/2)] = [0, 0, 0].

Links

Table of n, a(n) for n=1..70.

Example

a(5) = 2 (radius = 5*2-1 = 9): [15/2, 1/2, 7*sqrt(1/2)] and [8, 3, 4*sqrt(1/2)], because this is the minimal set of points whose cuboctahedral rotations and reflections comprise all of the primitive points in a fcc lattice that intersect a sphere of radius 9 centered on a point in the lattice. Additional points [9, 0, 0] and [15/2, 9/2, 3*sqrt(1/2)] are not counted as they are nonprimitive, reducing to [1, 0, 0] and [5/2, 3/2, sqrt(1/2)] respectively.

Prog

(PARI) a(n)={my(c=0, r=2*n-1); for(x=0, r, for(y=0, min(x, sqrtint(r^2-x^2)), for(o=0, 1, my(m=2*(r^2-(x+o/2)^2-(y+o/2)^2)); if(!issquare(m), next); my(z=sqrtint(m)); if(z>=0 && z%2==o && gcd([2*x+o, 2*y+o, z, r])==1, c+=if(y+o && (x-y==z || x+y+o==z), 2, 1))))); c/3}

Crossrefs

Cf. A387222, A386315.

Sequence in context: A241952 A236919 A138304 * A054860 A098745 A029158

Adjacent sequences: A387220 A387221 A387222 * A387224 A387225 A387226

Keyword

nonn

Author

Charles L. Hohn, Aug 22 2025

Status

approved