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A160849 Number of lines through at least 2 points of a 9 X n grid of points.

%I A160849 #12 Dec 25 2017 11:22:17
%S A160849 0,1,83,164,299,454,673,894,1183,1492,1855,2218,2653,3102,3623,4148,
%T A160849 4719,5310,5973,6638,7375,8124,8923,9730,10609,11502,12459,13424,
%U A160849 14443,15478,16585,17686,18859,20052,21299,22554,23869,25198,26599,28008
%N A160849 Number of lines through at least 2 points of a 9 X n grid of points.
%H A160849 Seiichi Manyama, <a href="/A160849/b160849.txt">Table of n, a(n) for n = 0..1000</a>
%H A160849 S. Mustonen, <a href="http://www.survo.fi/papers/PointsInGrid.pdf">On lines and their intersection points in a rectangular grid of points</a>
%F A160849 a(n) = (1/2)*(f(m,n,1) - f(m,n,2)) where f(m,n,k) = Sum((n-|kx|)*(m-|ky|)); -n < kx < n, -m < ky < m, (x,y)=1, m=9.
%F A160849 For another more efficient formula, see Mathematica code below.
%F A160849 Conjectures from _Colin Barker_, Dec 25 2017: (Start)
%F A160849 G.f.: x*(1 + 83*x + 165*x^2 + 382*x^3 + 618*x^4 + 971*x^5 + 1264*x^6 + 1607*x^7 + 1756*x^8 + 1873*x^9 + 1738*x^10 + 1571*x^11 + 1228*x^12 + 935*x^13 + 600*x^14 + 373*x^15 + 174*x^16 + 92*x^17 + 19*x^18 + 9*x^19 + 9*x^20) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
%F A160849 a(n) = -a(n-2) + a(n-5) + a(n-6) + 2*a(n-7) + a(n-8) + a(n-9) - a(n-11) - a(n-12) - 2*a(n-13) - a(n-14) - a(n-15) + a(n-18) + a(n-20) for n>21.
%F A160849 (End)
%t A160849 m=9;
%t A160849 a[0]=0; a[1]=1;
%t A160849 a[2]=m^2+2;
%t A160849 a[3]=2*m^2+3-Mod[m,2];
%t A160849 a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
%t A160849 p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
%t A160849 p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
%t A160849 p[i_]:=If[i>0,i,0]
%t A160849 p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
%t A160849 p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
%t A160849 p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
%t A160849 p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
%t A160849 p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
%t A160849 Table[a[n],{n,0,39}]
%Y A160849 Column k=9 of A295707.
%K A160849 nonn
%O A160849 0,3
%A A160849 _Seppo Mustonen_, May 28 2009