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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 (list; graph; refs; listen; history; text; internal format)
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=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

Adjacent sequences: A160842 A160843 A160844 * A160846 A160847 A160848

KEYWORD

nonn

AUTHOR

Seppo Mustonen, May 28 2009

STATUS

approved

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Last modified November 28 22:51 EST 2022. Contains 358421 sequences. (Running on oeis4.)