A160842 Number of lines through at least 2 points of a 2 X n grid of points.

0, 1, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502

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

Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = n^2 + 2 = A059100(n) for n > 1.

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

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

Sum_{n>=1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018

Mathematica

a[n_]:=If[n<2, n, n^2+2] Table[a[n], {n, 0, 50}]
Join[{0, 1}, Range[2, 50]^2+2] (* Harvey P. Dale, Feb 06 2015 *)

Prog

(PARI) Vec(-x*(2*x^3-4*x^2+3*x+1) / (x-1)^3 + O(x^100)) \\ Colin Barker, May 24 2015
(Magma) [0, 1] cat [n^2 + 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

Crossrefs

Cf. A295707, A107348.

Sequence in context: A073945 A083500 A102305 * A365351 A007745 A188556

Adjacent sequences: A160839 A160840 A160841 * A160843 A160844 A160845

Keyword

nonn,easy

Author

Seppo Mustonen, May 28 2009

Status

approved