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%I #69 Jan 01 2024 02:24:32
%S 0,3,13,40,108,275,681,1664,4040,9779,23637,57096,137876,332899,
%T 803729,1940416,4684624,11309731,27304157,65918120,159140476,
%U 384199155,927538873,2239276992,5406092952,13051462995,31509019045,76069501192,183648021540,443365544387
%N Number of 2 X n checkerboards (with at least one red square) in which the set of red squares is edge connected.
%C In other words, the number of connected (non-null) induced subgraphs in the n-ladder graph P_2 X P_n. - _Eric W. Weisstein_, May 02 2017
%C Also, the number of cycles in the grid graph P_3 X P_{n+1}. - _Andrew Howroyd_, Jun 12 2017
%H Vincenzo Librandi, <a href="/A059020/b059020.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InducedSubgraph.html">Induced Subgraph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,0,1).
%F a(n) = 2*a(n-1) + a(n-2) + 4*n - 1.
%F From _Jaume Oliver Lafont_, Nov 23 2008: (Start)
%F a(n) = 3*a(n-1) - a(n-2) - a(n-3) + 4;
%F a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4). (End)
%F G.f.: x*(3+x)/((1-2*x-x^2)*(1-x)^2). - _Jaume Oliver Lafont_, Sep 28 2009
%F Empirical observations (from Superseeker):
%F (1) if b(n) = a(n) + n then {b(n)} is A048777;
%F (2) if b(n) = a(n+3) - 3*a(n+2) - 3*a(n+1) + a(n) then {b(n)} is A052542;
%F (3) if b(n) = a(n+2) - 2*a(n+1) + a(n) then {b(n)} is A001333.
%F 4*a(n) = A002203(n+3) - 8*n - 14. - _Eric W. Weisstein_, May 02 2017
%F a(n) = 3*A048776(n-1) + A048776(n-2). - _R. J. Mathar_, May 12 2019
%F E.g.f.: (1/2)*exp(x)*(-7-4*x+7*cosh(sqrt(2)*x)+5*sqrt(2)*sinh(sqrt(2)*x)). - _Stefano Spezia_, Aug 25 2019
%t Join[{0},LinearRecurrence[{4, -4, 0, 1}, {3, 13, 40, 108}, 20]] (* _Eric W. Weisstein_, May 02 2017 *) (* adapted by _Vincenzo Librandi_, May 09 2017 *)
%t Table[(LucasL[n + 3, 2] - 8 n - 14)/4, {n, 0, 20}] (* _Eric W. Weisstein_, May 02 2017 *)
%o (Magma) I:=[0, 3, 13, 40];[n le 4 select I[n] else 4*Self(n-1) - 4*Self(n-2) + Self(n-4):n in [1..30]]; // _Marius A. Burtea_, Aug 25 2019
%Y Row 2 of A287151 and row 2 of A231829.
%Y See also A059021, A059524.
%Y Cf. A000129. - _Jaume Oliver Lafont_, Sep 28 2009
%Y Other sequences counting connected induced subgraphs: A020873, A059525, A286139, A286182, A286183, A286184, A286185, A286186, A286187, A286188, A286189, A286191, A285765, A285934, A286304.
%K nonn,easy
%O 0,2
%A _John W. Layman_, Dec 14 2000
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