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

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
• 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).

Crossrefs

Cf. A295707, A107348.

Programs

Magma

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

PARI

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

Formula

a(n) = n^2 + 2 = A059100(n) = A010000(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