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

0,3

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

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

Formula

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

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)

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 *)

Prog

(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
(PARI) 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

Crossrefs

4th row/column of A107348, A295707.

Sequence in context: A014640 A245587 A215137 * A256878 A285318 A040306

Adjacent sequences: A160841 A160842 A160843 * A160845 A160846 A160847

Keyword

nonn

Author

Seppo Mustonen, May 28 2009

Status

approved