

A222267


The number of distinct lines defined by an n X n X n grid of points.


6



28, 253, 1492, 5485, 17092, 41905, 95140, 191773, 364420, 638785, 1085500, 1745389, 2743084, 4136257, 6101740, 8747821, 12377764, 17066737, 23287564, 31174813, 41276548, 53767873, 69544324, 88722973, 112450132, 140859361, 175324636
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OFFSET

2,1


COMMENTS

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..n1]. 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


LINKS

Clive Tooth, Table of n, a(n) for n = 2..200 (using a method of Haukkanen & Merikoski) [Terms 2 through 70 were computed by Jon E. Schoenfield]
P. Haukkanen, J. K. Merikoski, Some formulas for numbers of line segments and lines in a rectangular grid, arXiv:1108.1041 [math.CO], 2011.
Clive Tooth, A C# class declaration


EXAMPLE

Each of the 28 pairs of points on a 2 X 2 X 2 grid of points defines a distinct line, so a(2) = 28.
Of the 351 pairs of points on a 3 X 3 X 3 grid, there are only 253 distinct lines, so a(3) = 253.


MATHEMATICA

mq[{x1_, y1_}, {x2_, y2_}] := If[x1 == x2, {x1}, {y2  y1, x2*y1  x1*y2}/(x2  x1)]; two[n_] := Block[{p = Tuples[Range@n, 2]},
Length@Union@Flatten[Table[mq[p[[i]], p[[j]]], {i, 2, n^2}, {j, i  1}], 1]]; coef[a_, b_] := Block[{d = b  a}, If[d[[1]] == 0, {0}, d *= Sign@d[[1]]/GCD @@ d; {a  d*a[[1]]/d[[1]], d}]]; a[n_] := Block[{p = Tuples[Range@n, 3]}, n*two[n]  1 + Length@Union@ Flatten[Table[coef[p[[i]], p[[j]]], {i, 2, n^3}, {j, i  1}], 1]]; Table[v = a[n]; Print@v; v, {n, 2, 12}] (* Giovanni Resta, Feb 14 2013 *)


CROSSREFS

Cf. A018808, A222268 (number of intersection points of these lines).
Sequence in context: A092341 A042522 A025517 * A240003 A225242 A189606
Adjacent sequences: A222264 A222265 A222266 * A222268 A222269 A222270


KEYWORD

nonn,nice


AUTHOR

Clive Tooth, Feb 13 2013


EXTENSIONS

a(6)a(12) from Giovanni Resta, Feb 14 2013
a(13)a(28) from Jon E. Schoenfield, Sep 16 2013


STATUS

approved



