A160850 - OEIS

%I #12 Dec 25 2017 11:22:26

%S 0,1,102,203,370,563,836,1111,1470,1855,2306,2757,3298,3857,4506,5159,

%T 5868,6603,7428,8255,9172,10105,11098,12101,13194,14305,15496,16697,

%U 17964,19251,20628,21997,23456,24941,26492,28053,29688,31341,33084

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

%H Seiichi Manyama, <a href="/A160850/b160850.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Mustonen, <a href="http://www.survo.fi/papers/PointsInGrid.pdf">On lines and their intersection points in a rectangular grid of points</a>

%F 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.

%F For another more efficient formula, see Mathematica code below.

%F Conjectures from _Colin Barker_, Dec 25 2017: (Start)

%F 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)).

%F 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.

%F (End)

%t m=10;

%t a[0]=0; a[1]=1;

%t a[2]=m^2+2;

%t a[3]=2*m^2+3-Mod[m,2];

%t a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]

%t p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]

%t p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]

%t p[i_]:=If[i>0,i,0]

%t p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]

%t p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]

%t p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]

%t p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]

%t p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]

%t Table[a[n],{n,0,39}]

%Y Column k=10 of A295707.

%K nonn

%O 0,3

%A _Seppo Mustonen_, May 28 2009