A160850 Number of lines through at least 2 points of a 10 X n grid of points.

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

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

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=10.

For another more efficient formula, see Mathematica code below.

Conjectures from Colin Barker, Dec 25 2017: (Start)

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)

Mathematica

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}]

Crossrefs

Column k=10 of A295707.

Sequence in context: A044715 A197816 A078787 * A202052 A284450 A173968

Adjacent sequences: A160847 A160848 A160849 * A160851 A160852 A160853

Keyword

nonn

Author

Seppo Mustonen, May 28 2009

Status

approved