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 A286184 Number of connected induced (non-null) subgraphs of the helm graph with 2n+1 nodes. 16
 6, 19, 56, 157, 430, 1171, 3204, 8857, 24794, 70303, 201712, 584677, 1708998, 5028715, 14873180, 44160817, 131499442, 392401207, 1172747208, 3508804477, 10506490526, 31477528579, 94344505396, 282848966857, 848161024650, 2543677767631, 7629355581344 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 Andrew Howroyd, Combinatorial Proof of Formula Eric Weisstein's World of Mathematics, Helm Graph Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph Index entries for linear recurrences with constant coefficients, signature (9,-31,51,-40,12). FORMULA a(n) = 3^n + (1 + n)*2^n - n. a(n) = 9*a(n-1) - 31*a(n-2) + 51*a(n-3) - 40*a(n-4) + 12*a(n-5). - Eric W. Weisstein, May 28 2017 G.f.: x*(6 - 35*x + 71*x^2 - 64*x^3 + 24*x^4)/((1 - 3*x)*(1 - 2*x)^2*(1 - x)^2). - Vincenzo Librandi, May 21 2017 E.g.f.: exp(3*x) - x*exp(x) + exp(2*x)*(1 + 2*x) - 2. - Stefano Spezia, Aug 25 2022 MATHEMATICA a[n_] := Block[{g = Graph@ Flatten@ Table[{i <-> Mod[i, n] + 1, i <-> n + Mod[i, n] + 1, i <-> 2 n + 1}, {i, n}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[2 n + 1]}]]; Array[a, 8] Table[3^n + (1 + n) 2^n - n, {n, 30}] (* Vincenzo Librandi, May 21 2017 *) CoefficientList[Series[(6 - 35 x + 71 x^2 - 64 x^3 + 24 x^4) / ((1-3x)(1-2x)^2(1-x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 21 2017 *) LinearRecurrence[{9, -31, 51, -40, 12}, {6, 19, 56, 157, 430}, 20] (* Eric W. Weisstein, May 28 2017 *) PROG (Magma) [3^n + (1+n)*2^n - n: n in [1..30]]; // Vincenzo Librandi, May 21 2017 CROSSREFS Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen). Sequence in context: A272562 A176883 A274599 * A027044 A057571 A238055 Adjacent sequences: A286181 A286182 A286183 * A286185 A286186 A286187 KEYWORD nonn,easy AUTHOR Giovanni Resta, May 04 2017 EXTENSIONS a(17)-a(27) from Andrew Howroyd, May 21 2017 STATUS approved

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Last modified July 20 14:47 EDT 2024. Contains 374459 sequences. (Running on oeis4.)