A160843 Number of lines through at least 2 points of a 3 X n grid of points.

0, 1, 11, 20, 35, 52, 75, 100, 131, 164, 203, 244, 291, 340, 395, 452, 515, 580, 651, 724, 803, 884, 971, 1060, 1155, 1252, 1355, 1460, 1571, 1684, 1803, 1924, 2051, 2180, 2315, 2452, 2595, 2740, 2891, 3044, 3203, 3364, 3531, 3700, 3875, 4052, 4235, 4420

Offset: 0

Author

Seppo Mustonen, May 28 2009

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• S. Mustonen, On lines and their intersection points in a rectangular grid of points
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

3rd row/column of A107348, A295707.

Programs

Magma

[0, 1] cat [2*n^2 + 3 - n mod 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

Mathematica

a[n_]:=If[n<2,n,2*n^2+3-Mod[n,2]] Table[a[n],{n,0,47}]
Join[{0, 1}, LinearRecurrence[{2, 0, -2, 1}, {11, 20, 35, 52}, 20]] (* G. C. Greubel, Apr 30 2018 *)

PARI

Vec(-x*(3*x^4-3*x^3-2*x^2+9*x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, May 24 2015

Formula

a(n) = 2*n^2 + 3 - n mod 2.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 5. - Colin Barker, May 24 2015

G.f.: -x*(3*x^4 - 3*x^3 - 2*x^2 + 9*x + 1) / ((x-1)^3*(x+1)). - Colin Barker, May 24 2015