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A018808
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Number of lines through at least 2 points of an n X n grid of points.
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17
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0, 0, 6, 20, 62, 140, 306, 536, 938, 1492, 2306, 3296, 4722, 6460, 8830, 11568, 14946, 18900, 23926, 29544, 36510, 44388, 53586, 63648, 75674, 88948, 104374, 121032, 139966, 160636, 184466, 209944, 239050, 270588, 305478, 342480, 383370, 427020
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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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(1/2) * (f(n, 1) - f(n, 2)) where f(n, k) = Sum ((n - |x|)(n - |y|)); -n < x < n, -n < y < n, (x, y)=k.
(1/2) * (f(n, 1) - f(n, 2)) where f(n, k) = Sum ((n - |kx|)(n - |ky|)); -n < kx < n, -n < ky < n, (x, y)=1. - Seppo Mustonen, Apr 18 2009
a(0) = L(0,1) = R1(0) = 0, a(n) = L(n,n) = 2L(n-1,n) - L(n-1,n-1) + R1(n), L(n-1,n) = 2L(n-1,n-1) - L(n-2,n-1) + R2(n), R1(n) = R1(n-1) + 4(phi(n-1) - e(n)), e(n)=0, n even, e(n) = phi((n-1)/2), n odd, R2(n) = (n-1)phi(n-1), n even, R2(n)=(n-1)phi(n-1)/2, n=1 mod 4, R2(n)=0, n=3 mod 4. - Seppo Mustonen, Apr 25 2009
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MATHEMATICA
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L[0]=0; L1[1]=0; R1[1]=0;
L[n_]:=L[n]=2*L1[n]-L[n-1]+R1[n]
L1[n_]:=L1[n]=2*L[n-1]-L1[n-1]+R2[n]
R1[n_]:=R1[n]=R1[n-1]+4*(EulerPhi[n-1]-e[n])
e[n_]:=If[Mod[n, 2]==0, 0, EulerPhi[(n-1)/2]]
R2[n_]:= If[Mod[n, 2]==0, (n-1)*EulerPhi[n-1], If[Mod[n, 4]==1, (n-1)*EulerPhi[n-1]/2, 0]]
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CROSSREFS
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Cf. A222267 (lines defined by n X n X n grid of points).
A288187 is the main entry for these graphs.
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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