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A160850
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Number of lines through at least 2 points of a 10 X n grid of points.
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3
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0, 1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306, 2757, 3298, 3857, 4506, 5159, 5868, 6603, 7428, 8255, 9172, 10105, 11098, 12101, 13194, 14305, 15496, 16697, 17964, 19251, 20628, 21997, 23456, 24941, 26492, 28053, 29688, 31341, 33084
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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=10.
For another more efficient formula, see Mathematica code below.
G.f.: x*(1 + 102*x + 204*x^2 + 472*x^3 + 766*x^4 + 1205*x^5 + 1571*x^6 + 1999*x^7 + 2188*x^8 + 2334*x^9 + 2168*x^10 + 1959*x^11 + 1531*x^12 + 1165*x^13 + 746*x^14 + 462*x^15 + 214*x^16 + 112*x^17 + 21*x^18 + 10*x^19 + 10*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=10;
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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