A177720 Number of line segments connecting exactly 4 points in an n x n grid of points.

0, 0, 0, 10, 28, 54, 104, 170, 252, 394, 568, 774, 1068, 1410, 1800, 2374, 3028, 3762, 4656, 5646, 6732, 8190, 9792, 11538, 13636, 15910, 18360, 21334, 24532, 27954, 31856, 36014, 40428, 45798, 51504, 57546, 64228, 71278, 78696, 87466, 96700

Offset: 1

Author

Seppo Mustonen, May 13 2010

Keywords

• nonn

Comments

a(n) is also the number of pairs of points visible to each other exactly through 2 points in an n x n grid of points.

Links

• S. Mustonen, On lines going through a given number of points in a rectangular grid of points
• Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points [Local copy]

Programs

Mathematica

j=3;
a[n_]:=a[n]=If[n<=j,0,2*a1[n]-a[n-1]+R1[n]]
a1[n_]:=a1[n]=If[n<=j,0,2*a[n-1]-a1[n-1]+R2[n]]
R1[n_]:=R1[n]=If[n<=j,0,R1[n-1]+4*S[n]]
R2[n_]:=(n-1)*S[n]
S[n_]:=If[Mod[n-1,j]==0,EulerPhi[(n-1)/j],0]
Table[a[n],{n,1,50}]

Formula

See Mathematica code.