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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
Eric Weisstein's World of Mathematics, Helm Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph
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
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.)