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

0,3

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).

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

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 *)

Prog

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

Crossrefs

3rd row/column of A107348, A295707.

Sequence in context: A109376 A100038 A370917 * A153368 A068600 A158235

Adjacent sequences: A160840 A160841 A160842 * A160844 A160845 A160846

Keyword

nonn,easy

Author

Seppo Mustonen, May 28 2009

Status

approved