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A160848 Number of lines through at least 2 points of an 8 X n grid of points 0
0, 1, 66, 131, 238, 361, 534, 709, 938, 1183, 1470, 1759, 2104, 2459, 2870, 3287, 3740, 4209, 4734, 5261, 5844, 6437, 7070, 7711, 8408, 9115, 9872, 10637, 11444, 12265, 13142, 14015, 14944, 15889, 16876, 17871, 18914, 19967, 21076, 22193, 23352 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..40.

S. Mustonen, On lines and their intersection points in a rectangular grid of points

FORMULA

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=8

For another more efficient formula, see Mathematica code below.

MATHEMATICA

m=8;

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, 40}]

CROSSREFS

Sequence in context: A044189 A044570 A118163 * A160278 A206030 A174929

Adjacent sequences:  A160845 A160846 A160847 * A160849 A160850 A160851

KEYWORD

nonn

AUTHOR

Seppo Mustonen, May 28 2009

STATUS

approved

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Last modified August 20 11:27 EDT 2017. Contains 290835 sequences.