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%I #22 Apr 24 2023 15:20:34
%S 6,26,76,218,664,2174,7452,26130,92512,328774,1170052,4166106,
%T 14836488,52839374,188188396,670240802,2387095600,8501764310,
%U 30279479508,107841961962,384084839128,1367938434910,4871984975932,17351831789874,61799465313024
%N Number of connected induced (non-null) subgraphs of the gear graph with 2n+1 nodes.
%H Andrew Howroyd, <a href="/A286188/b286188.txt">Table of n, a(n) for n = 1..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>
%F a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3) + 3*a(n-4) - 2*a(n-5), for n>5.
%F a(n) = A206776(n) + 4*n^2 - 2*n + 1. - _Eric W. Weisstein_, May 08 2017
%F G.f.: 2*x*(3 - 5*x - 10*x^2 - x^3 - 3*x^4) / ((1 - x)^3*(1 - 3*x - 2*x^2)). - _Colin Barker_, May 31 2017
%t a[n_] := Block[{g = Graph@ Flatten[{Table[i <-> 2 n + 1, {i, 2, 2 n, 2}], Table[i <-> Mod[i, 2 n] + 1, {i, 2 n}]}]}, -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[2 n + 1]}]]; Array[a, 8]
%Y Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A286189 (rook), A285765 (queen).
%K nonn
%O 1,1
%A _Giovanni Resta_, May 04 2017
%E a(16)-a(25) from _Andrew Howroyd_, May 20 2017
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