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A160845
Number of lines through at least 2 points of a 5 X n grid of points.
2
0, 1, 27, 52, 93, 140, 207, 274, 361, 454, 563, 676, 809, 944, 1099, 1258, 1433, 1614, 1815, 2016, 2237, 2464, 2707, 2954, 3221, 3490, 3779, 4072, 4381, 4696, 5031, 5366, 5721, 6082, 6459, 6840, 7241, 7644, 8067, 8494, 8937, 9386, 9855, 10324, 10813
OFFSET
0,3
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=5.
For another more efficient formula, see Mathematica code below.
Conjectures from Colin Barker, May 24 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-5) - a(n-7) + a(n-8) for n > 7.
G.f.: x*(5*x^8 + x^6 + 16*x^5 + 20*x^4 + 40*x^3 + 25*x^2 + 26*x + 1) / ((1 - x)^3*(x + 1)*(x^2 + 1)*(x^2 + x + 1)).
(End)
MATHEMATICA
m=5;
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, 44}]
CROSSREFS
5th row/column of A107348, A295707.
Sequence in context: A265683 A044079 A044460 * A216224 A255364 A082915
KEYWORD
nonn
AUTHOR
Seppo Mustonen, May 28 2009
STATUS
approved