cp's OEIS Frontend

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.

%I A387222 #5 Aug 31 2025 17:57:42
%S A387222 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,
%T A387222 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,
%U A387222 7,17,9,22,1,29,11,18,6,20,17,19,4,20,11,30,6,26
%N 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.
%C A387222 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).
%C A387222 Odd n with nonprimitive points removed gives A387223.
%e A387222 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.
%o A387222 (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}
%Y A387222 Cf. A386315, A387223.
%K A387222 nonn,new
%O A387222 0,4
%A A387222 _Charles L. Hohn_, Aug 22 2025