A387222 Number of noncongruent points in a face-centered cubic lattice that intersect a sphere of radius n centered on a point in the lattice.

1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 4, 2, 5, 3, 7, 1, 6, 4, 6, 3, 8, 4, 7, 2, 11, 5, 9, 3, 9, 7, 9, 1, 11, 6, 17, 4, 11, 6, 13, 3, 12, 8, 12, 4, 18, 7, 13, 2, 17, 11, 16, 5, 15, 9, 21, 3, 17, 9, 16, 7, 17, 9, 22, 1, 29, 11, 18, 6, 20, 17, 19, 4, 20, 11, 30, 6, 26

Offset

0,4

Comments

Here congruence is relative to the 48-point cuboctahedral symmetry in a fcc lattice. The symmetric rotations and reflections of the points that comprise a(n), with redundancies removed for points that lie on axis planes, gives A386315(n).

Odd n with nonprimitive points removed gives A387223.

Links

Table of n, a(n) for n=0..77.

Example

a(5) = 3: [5, 0, 0], [4, 3, 0], and [4, 1, 4*sqrt(1/2)], because this is the minimal set of points whose cuboctahedral rotations and reflections comprise all of the points in a fcc lattice that intersect a sphere of radius 5 centered on a point in the lattice.

Prog

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

Crossrefs

Cf. A386315, A387223.

Sequence in context: A070940 A020651 A294442 * A281392 A287051 A368147

Adjacent sequences: A387219 A387220 A387221 * A387223 A387224 A387225

Keyword

nonn

Author

Charles L. Hohn, Aug 22 2025

Status

approved