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Number of intersections between a sphere inscribed in a cube and the n X n X n cubes resulting from a cubic lattice subdivision of the enclosing cube.
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%I #23 Jun 18 2022 23:02:41

%S 8,26,56,98,152,194,272,362,440,530,656,746,872,1034,1160,1298,1496,

%T 1658,1856,1994,2240,2450,2624,2906,3128,3362,3656,3890,4208,4442,

%U 4760,5090,5360,5714,6032,6362,6752,7106,7496,7826,8216,8618,9080,9458,9896

%N Number of intersections between a sphere inscribed in a cube and the n X n X n cubes resulting from a cubic lattice subdivision of the enclosing cube.

%C A concise description of the problem is given by Clive Tooth in the Seaman, Tooth link. Sequence terms up to n=10 were first given by Dave Seaman. Cubes having at least one vertex on the sphere and all other vertices either all inside or all outside the sphere are counted as 1/2. a(n) is asymptotic to (3/2)*Pi*n^2. (Clive Tooth) The terms a(2),...,a(6) are identical with A005897(n-1) (points on surface of cube with square grid on its faces).

%H Hugo Pfoertner, <a href="/A085690/b085690.txt">Table of n, a(n) for n = 2..1000</a>

%H Hugo Pfoertner, <a href="/A085690/a085690_1.f.txt">FORTRAN program to count intersections.</a>

%H Dave Seaman, Clive Tooth, <a href="http://groups.google.com/group/sci.math/browse_thread/thread/590055b59dc3d406/6bf957c1724061fb">Sphere/Cube Intersections.</a> Discussion in Newsgroup sci.math.

%e a(2)=8 because all 8 cubes resulting from a 2*2*2 subdivision of a cube are intersected by a sphere inscribed in the large cube.

%e a(4)=56: 8 central cubes of 4*4*4=64 not intersected.

%o FORTRAN and C# programs are given at the links.

%Y Cf. A005897, A008574.

%K nonn

%O 2,1

%A _Hugo Pfoertner_, Jul 17 2003

%E Corrected overflow in program and b-file by _Hugo Pfoertner_, Apr 09 2016