A160844 Number of lines through at least 2 points of a 4 X n grid of points.

0, 1, 18, 35, 62, 93, 136, 181, 238, 299, 370, 445, 532, 621, 722, 827, 942, 1061, 1192, 1325, 1470, 1619, 1778, 1941, 2116, 2293, 2482, 2675, 2878, 3085, 3304, 3525, 3758, 3995, 4242, 4493, 4756, 5021, 5298, 5579, 5870, 6165, 6472, 6781, 7102, 7427, 7762

Offset: 0

Author

Seppo Mustonen, May 28 2009

Keywords

• nonn

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
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

4th row/column of A107348, A295707.

Programs

Magma

I:=[18, 35, 62, 93, 136, 181]; [0,1] cat [n le 6 select I[n] else Self(n-1) +Self(n-2) -Self(n-4) -Self(n-5) +Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 30 2018

Mathematica

a[0]=0; a[1]=1; a[2]=18; a[3]=35; a[n_]:=a[n]=a[n]=2*a[n-1]-a[n-2]+R[n] c4={10,4,12,2,12,4}; R[n_]:=c4[[Mod[n+2,6]+1]] Table[a[n],{n,0,46}]
Join[{0,1}, LinearRecurrence[{1,1,0,-1,-1,1}, {18, 35, 62, 93, 136, 181}, 50]] (* G. C. Greubel, Apr 30 2018 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x*(4*x^6-3*x^4+9*x^3+16*x^2+ 17*x+1 )/((1-x)^3*(x+1)*(x^2+x+1)))) \\ G. C. Greubel, Apr 30 2018

Formula

a(n) = 2*a(n-1) - a(n-2) + C(mod(n+2,6) + 1), C=(10,4,12,2,12,4), for n >= 4.

From Colin Barker, May 24 2015: (Start)

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 5.

G.f.: x*(4*x^6 - 3*x^4 + 9*x^3 + 16*x^2 + 17*x + 1) / ((1-x)^3*(x + 1)*(x^2 + x + 1)).

(End)