|
Search: id:A160847
|
|
|
| A160847 |
|
Number of lines through at least 2 points of an 7 X n grid of points |
|
+0
1
|
|
| 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; listen)
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
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.
|
|
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: A075894 A044140 A044521 this_sequence A015705 A044238 A044619
Adjacent sequences: A160844 A160845 A160846 this_sequence A160848 A160849 A160850
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Seppo Mustonen (seppo.mustonen(AT)helsinki.fi), May 28 2009
|
|
|
Search completed in 0.004 seconds
|
