A386315 Number of points in a face-centered cubic lattice that intersect a sphere of radius n centered on a point in the lattice.

1, 12, 12, 36, 12, 84, 36, 108, 12, 108, 84, 132, 36, 180, 108, 252, 12, 204, 108, 228, 84, 324, 132, 300, 36, 444, 180, 324, 108, 372, 252, 396, 12, 396, 204, 756, 108, 468, 228, 540, 84, 492, 324, 516, 132, 756, 300, 588, 36, 780, 444, 612, 180, 660, 324

Offset: 0

Author

Charles L. Hohn, Aug 15 2025

Keywords

• nonn
• new

Comments

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.

Examples

			a(3) = 36, which is the sum of 4 90-degree rotations of [3, 0, 0], 8 90-degree rotations and vertical reflections of [3/2, 3/2, 3*sqrt(1/2)] and [1, 0, 4*sqrt(1/2)], and 16 90-degree rotations and vertical and horizontal reflections of [5/2, 3/2, sqrt(1/2)].
		

Links

• Charles L. Hohn, a(0) to a(15), animated - see Comments

Crossrefs

Cf. A004015, A016725.

Cf. A387222, A387223, A278081.

Programs

PARI

a(n)={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+=2^(4-if(!z, 1)-if(x==y, 1)-if(!min(x, y) && !o, 1)-if(!vecmax([x, y, z, o]), 1)))))); c}

PARI

a(n)={if(!n, return(1)); my(f=Vec(factor(n)), o=12, r=o); for(i=if(#f[1] && f[1][1]==2, 2, 1), #f[1], my(m=if(f[1][i]%8>=4, 2)); f[2][i]++; while(f[2][i]--, o=o*f[1][i]+r*m); r=o); o}

Formula

a(n) = A004015(n^2).

a(2*n) = a(n).

a(p*n) = p*a(n) where p is a prime and p mod 8 is in {1, 3}.

a(p*n) = p*a(n) + 2*a(n/p^c) where p is a prime, p mod 8 is in {5, 7}, and c is the count of prime factors p in n.