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Number of line segments connecting exactly 8 points in an n x n grid of points
0

%I #2 Mar 30 2012 18:59:35

%S 0,0,0,0,0,0,0,18,44,78,120,170,228,294,400,522,660,814,984,1170,1372,

%T 1682,2024,2398,2804,3242,3712,4214,4868,5570,6320,7118,7964,8858,

%U 9800,11086,12452,13898,15424,17030,18716,20482,22504,24622,26836,29146

%N Number of line segments connecting exactly 8 points in an n x n grid of points

%C a(n) is also the number of pairs of points visible to each other exactly through 6 points in an n x n grid of points.

%H S. Mustonen, <a href="http://www.survo.fi/papers/LinesInGrid2.pdf">On lines going through a given number of points in a rectangular grid of points</a>

%F See Mathematica code.

%t j=7;

%t a[n_]:=a[n]=If[n<=j,0,2*a1[n]-a[n-1]+R1[n]]

%t a1[n_]:=a1[n]=If[n<=j,0,2*a[n-1]-a1[n-1]+R2[n]]

%t R1[n_]:=R1[n]=If[n<=j,0,R1[n-1]+4*S[n]]

%t R2[n_]:=(n-1)*S[n]

%t S[n_]:=If[Mod[n-1,j]==0,EulerPhi[(n-1)/j],0]

%t Table[a[n],{n,1,50}]

%K nonn

%O 1,8

%A _Seppo Mustonen_, May 13 2010