A085690 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.

8, 26, 56, 98, 152, 194, 272, 362, 440, 530, 656, 746, 872, 1034, 1160, 1298, 1496, 1658, 1856, 1994, 2240, 2450, 2624, 2906, 3128, 3362, 3656, 3890, 4208, 4442, 4760, 5090, 5360, 5714, 6032, 6362, 6752, 7106, 7496, 7826, 8216, 8618, 9080, 9458, 9896

Offset

2,1

Comments

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).

Links

Hugo Pfoertner, Table of n, a(n) for n = 2..1000

Hugo Pfoertner, FORTRAN program to count intersections.

Dave Seaman, Clive Tooth, Sphere/Cube Intersections. Discussion in Newsgroup sci.math.

Example

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

Prog

FORTRAN and C# programs are given at the links.

Crossrefs

Cf. A005897, A008574.

Sequence in context: A126264 A347677 A225274 * A005897 A215097 A331242

Adjacent sequences: A085687 A085688 A085689 * A085691 A085692 A085693

Keyword

nonn

Author

Hugo Pfoertner, Jul 17 2003

Extensions

Corrected overflow in program and b-file by Hugo Pfoertner, Apr 09 2016

Status

approved