0,3

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

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=9.

For another more efficient formula, see Mathematica code below.

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

G.f.: x*(1 + 83*x + 165*x^2 + 382*x^3 + 618*x^4 + 971*x^5 + 1264*x^6 + 1607*x^7 + 1756*x^8 + 1873*x^9 + 1738*x^10 + 1571*x^11 + 1228*x^12 + 935*x^13 + 600*x^14 + 373*x^15 + 174*x^16 + 92*x^17 + 19*x^18 + 9*x^19 + 9*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)

m=9;

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}]

Column k=9 of A295707.

Sequence in context: A142118 A044253 A044634 * A277807 A246874 A136079

Adjacent sequences: A160846 A160847 A160848 * A160850 A160851 A160852

Seppo Mustonen, May 28 2009

approved