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A160849
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Number of lines through at least 2 points of a 9 X n grid of points.
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2
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0, 1, 83, 164, 299, 454, 673, 894, 1183, 1492, 1855, 2218, 2653, 3102, 3623, 4148, 4719, 5310, 5973, 6638, 7375, 8124, 8923, 9730, 10609, 11502, 12459, 13424, 14443, 15478, 16585, 17686, 18859, 20052, 21299, 22554, 23869, 25198, 26599, 28008
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OFFSET
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0,3
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LINKS
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FORMULA
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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.
For another more efficient formula, see Mathematica code below.
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)).
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.
(End)
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MATHEMATICA
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m=9;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m, 2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m, n]+2*p4[m, n]
p1[m_, n_]:=Sum[p2[m, n, y], {y, 1, m-1}]
p2[m_, n_, y_]:=If[GCD[y, n-1]==1, m-y, 0]
p[i_]:=If[i>0, i, 0]
p2[m_, n_, x_, y_]:=p2[m, n, x, y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_, n_, x_, y_]:=p2[m, n, x, y]-2*p2[m, n-1, x, y]+p2[m, n-2, x, y]
p4[m_, n_]:=p4[m, n]=If[Mod[n, 2]==0, 0, p42[m, n]]
p42[m_, n_]:=p42[m, n]=Sum[p43[m, n, y], {y, 1, m-1}]
p43[m_, n_, y_]:=If[GCD[(n-1)/2, y]==1, p3[m, n, (n-1)/2, y], 0]
Table[a[n], {n, 0, 39}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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