cp's OEIS Frontend

Given a cubic n X n X n grid of points, a collection of lines is produced by constructing a line through every pair of points. A222267 gives the count of such lines. A collection of points is produced by taking the points of intersection of those lines. A222268 gives the count of such points. Each point in such a collection will lie on a particular sphere centered at the center of the cubic grid. a(n) is the total number of such spheres, including the radius-zero sphere.

Given a cubic n X n X n grid of points, a(n) is the number of distinct lines produced by constructing a line through every pair of points.

Define the grid as consisting of the set of n^3 distinct points whose x, y and z coordinates are all integers in [0..n-1]. Assign to each grid point a distinct index j = x + n*y + n^2*z. For each pair of grid points P_A and P_B (where P_A is the one with the lower index j), let L be the line that passes through both grid points, and let S be the segment of that line from P_A to P_B. Examine each of the C(n^3,2) pairs of distinct grid points P_A and P_B; a(n) is the number of those pairs for which S does not pass through any other grid points between P_A and P_B, nor does L pass through any other grid points beyond the P_A end of S. - Jon E. Schoenfield, Sep 21 2013

Conjecture: a(n) is approximately 0.3639537*n^6, with a relative error of about 10^-5 when n is near 200. - Clive Tooth, Mar 03 2016