%I #34 Sep 08 2022 08:45:45

%S 0,1,6,11,18,27,38,51,66,83,102,123,146,171,198,227,258,291,326,363,

%T 402,443,486,531,578,627,678,731,786,843,902,963,1026,1091,1158,1227,

%U 1298,1371,1446,1523,1602,1683,1766,1851,1938,2027,2118,2211,2306,2403,2502

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

%H Vincenzo Librandi, <a href="/A160842/b160842.txt">Table of n, a(n) for n = 0..10000</a>

%H S. Mustonen, <a href="http://www.survo.fi/papers/PointsInGrid.pdf">On lines and their intersection points in a rectangular grid of points</a>

%H Seppo Mustonen, <a href="/A018808/a018808.pdf">On lines and their intersection points in a rectangular grid of points</a> [Local copy]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. - _Colin Barker_, May 24 2015

%F G.f.: -x*(2*x^3 - 4*x^2 + 3*x + 1) / (x-1)^3. - _Colin Barker_, May 24 2015

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

%t a[n_]:=If[n<2,n,n^2+2] Table[a[n],{n,0,50}]

%t Join[{0,1},Range[2,50]^2+2] (* _Harvey P. Dale_, Feb 06 2015 *)

%o (PARI) Vec(-x*(2*x^3-4*x^2+3*x+1) / (x-1)^3 + O(x^100)) \\ _Colin Barker_, May 24 2015

%o (Magma) [0,1] cat [n^2 + 2: n in [2..100]]; // _G. C. Greubel_, Apr 30 2018

%Y Cf. A295707, A107348.

%K nonn,easy

%O 0,3

%A _Seppo Mustonen_, May 28 2009