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A160847 Number of lines through at least 2 points of an 7 X n grid of points. 0
0, 1, 51, 100, 181, 274, 405, 536, 709, 894, 1111, 1330, 1591, 1858, 2167, 2482, 2825, 3180, 3577, 3974, 4413, 4860, 5339, 5824, 6351, 6884, 7455, 8032, 8641, 9262, 9925, 10584, 11285, 11998, 12743, 13494, 14283, 15078, 15915, 16758, 17633, 18516 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..41.

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=7.

For another more efficient formula, see Mathematica code below.

Empirical g.f.: -x*(7*x^14 +8*x^12 +43*x^11 +50*x^10 +117*x^9 +135*x^8 +204*x^7 +173*x^6 +211*x^5 +142*x^4 +131*x^3 +50*x^2 +50*x +1) / ((x -1)^3*(x +1)*(x^2 -x +1)*(x^2 +1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, May 24 2015

MATHEMATICA

m=7;

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, 41}]

CROSSREFS

Sequence in context: A229274 A044140 A044521 * A260517 A235878 A015705

Adjacent sequences:  A160844 A160845 A160846 * A160848 A160849 A160850

KEYWORD

nonn

AUTHOR

Seppo Mustonen, May 28 2009

STATUS

approved

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Last modified May 29 21:21 EDT 2017. Contains 287257 sequences.