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 A059020 Number of 2 X n checkerboards (with at least one red square) in which the set of red squares is edge connected. 21
 0, 3, 13, 40, 108, 275, 681, 1664, 4040, 9779, 23637, 57096, 137876, 332899, 803729, 1940416, 4684624, 11309731, 27304157, 65918120, 159140476, 384199155, 927538873, 2239276992, 5406092952, 13051462995, 31509019045, 76069501192, 183648021540, 443365544387 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 Also, the number of cycles in the grid graph P_3 X P_{n+1}. - Andrew Howroyd, Jun 12 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Connected Graph. Eric Weisstein's World of Mathematics, Induced Subgraph. Eric Weisstein's World of Mathematics, Ladder Graph. Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1). FORMULA a(n) = 2*a(n-1) + a(n-2) + 4*n - 1. From Jaume Oliver Lafont, Nov 23 2008: (Start) a(n) = 3*a(n-1) - a(n-2) - a(n-3) + 4; a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4). (End) G.f.: x*(3+x)/((1-2*x-x^2)*(1-x)^2). - Jaume Oliver Lafont, Sep 28 2009 Empirical observations (from Superseeker): (1) if b(n) = a(n) + n then {b(n)} is A048777; (2) if b(n) = a(n+3) - 3*a(n+2) - 3*a(n+1) + a(n) then {b(n)} is A052542; (3) if b(n) = a(n+2) - 2*a(n+1) + a(n) then {b(n)} is A001333. 4*a(n) = A002203(n+3) - 8*n - 14. - Eric W. Weisstein, May 02 2017 a(n) = 3*A048776(n-1) + A048776(n-2). - R. J. Mathar, May 12 2019 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 MATHEMATICA 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 *) Table[(LucasL[n + 3, 2] - 8 n - 14)/4, {n, 0, 20}] (* Eric W. Weisstein, May 02 2017 *) PROG (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 CROSSREFS Row 2 of A287151 and row 2 of A231829. See also A059021, A059524. Cf. A000129. - Jaume Oliver Lafont, Sep 28 2009 Other sequences counting connected induced subgraphs: A020873, A059525, A286139, A286182, A286183, A286184, A286185, A286186, A286187, A286188, A286189, A286191, A285765, A285934, A286304. Sequence in context: A147042 A018492 A227446 * A290720 A289654 A360787 Adjacent sequences: A059017 A059018 A059019 * A059021 A059022 A059023 KEYWORD nonn,easy AUTHOR John W. Layman, Dec 14 2000 STATUS approved

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