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%I #27 Mar 19 2021 09:56:49
%S 0,1,27,52,93,140,207,274,361,454,563,676,809,944,1099,1258,1433,1614,
%T 1815,2016,2237,2464,2707,2954,3221,3490,3779,4072,4381,4696,5031,
%U 5366,5721,6082,6459,6840,7241,7644,8067,8494,8937,9386,9855,10324,10813
%N Number of lines through at least 2 points of a 5 X n grid of points.
%H Seiichi Manyama, <a href="/A160845/b160845.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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=5.
%F For another more efficient formula, see Mathematica code below.
%F Conjectures from _Colin Barker_, May 24 2015: (Start)
%F a(n) = a(n-1) + a(n-3) - a(n-5) - a(n-7) + a(n-8) for n > 7.
%F G.f.: x*(5*x^8 + x^6 + 16*x^5 + 20*x^4 + 40*x^3 + 25*x^2 + 26*x + 1) / ((1 - x)^3*(x + 1)*(x^2 + 1)*(x^2 + x + 1)).
%F (End)
%t m=5;
%t a[0]=0; a[1]=1;
%t a[2]=m^2+2;
%t a[3]=2*m^2+3-Mod[m,2];
%t a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
%t p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
%t p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
%t p[i_]:=If[i>0,i,0]
%t p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
%t p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
%t p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
%t p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
%t p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
%t Table[a[n],{n,0,44}]
%Y 5th row/column of A107348, A295707.
%K nonn
%O 0,3
%A _Seppo Mustonen_, May 28 2009
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