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A286186 Number of connected induced (non-null) subgraphs of the friendship graph with 2n+1 nodes.

%I A286186 #19 Feb 16 2025 08:33:44
%S A286186 7,22,73,268,1039,4114,16405,65560,262171,1048606,4194337,16777252,
%T A286186 67108903,268435498,1073741869,4294967344,17179869235,68719476790,
%U A286186 274877907001,1099511627836,4398046511167,17592186044482,70368744177733,281474976710728,1125899906842699
%N A286186 Number of connected induced (non-null) subgraphs of the friendship graph with 2n+1 nodes.
%H A286186 Colin Barker, <a href="/A286186/b286186.txt">Table of n, a(n) for n = 1..1000</a>
%H A286186 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DutchWindmillGraph.html">Dutch Windmill Graph</a>
%H A286186 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>
%H A286186 Wikipedia, <a href="https://en.wikipedia.org/wiki/Friendship_graph">Friendship graph</a>
%H A286186 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,4).
%F A286186 a(n) = 4^n + 3*n.
%F A286186 From _Colin Barker_, May 21 2017: (Start)
%F A286186 G.f.: x*(7 - 20*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)).
%F A286186 a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3) for n>3. (End)
%F A286186 E.g.f.: exp(x)*(exp(3*x) + 3*x) - 1. - _Stefano Spezia_, Aug 25 2022
%t A286186 Table[4^n + 3 n, {n, 30}]
%t A286186 LinearRecurrence[{6,-9,4},{7,22,73},40] (* _Harvey P. Dale_, May 25 2019 *)
%o A286186 (PARI) Vec(x*(7 - 20*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)) + O(x^30)) \\ _Colin Barker_, May 21 2017
%Y A286186 Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).
%K A286186 nonn,easy
%O A286186 1,1
%A A286186 _Giovanni Resta_, May 04 2017