login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A160846 Number of lines through at least 2 points of a 6 X n grid of points. 2
0, 1, 38, 75, 136, 207, 306, 405, 534, 673, 836, 1003, 1200, 1401, 1632, 1869, 2128, 2397, 2696, 2995, 3324, 3661, 4022, 4389, 4786, 5187, 5616, 6051, 6510, 6979, 7478, 7975, 8502, 9039, 9600, 10167, 10762, 11361, 11990, 12625, 13284, 13951, 14648 (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=6.

For another more efficient formula, see Mathematica code below.

Empirical g.f.: -x*(6*x^12 + 6*x^11 + 7*x^10 + 32*x^9 + 63*x^8 + 117*x^7 + 156*x^6 + 192*x^5 + 168*x^4 + 135*x^3 + 75*x^2 + 38*x + 1) / ((x - 1)^3*(x + 1)*(x^2 + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)). - Colin Barker, May 24 2015

MATHEMATICA

m=6;

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, 42}]

CROSSREFS

Sequence in context: A032506 A044104 A044485 * A219999 A219997 A039421

Adjacent sequences: A160843 A160844 A160845 * A160847 A160848 A160849

KEYWORD

nonn

AUTHOR

Seppo Mustonen, May 28 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 08:36 EST 2022. Contains 358493 sequences. (Running on oeis4.)