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%I #13 May 25 2017 09:05:31
%S 7,21,45,129,373,1177,3741,12065,38949,125993,407597,1318961,4268085,
%T 13811769,44695613,144638017,468058181,1514668105,4901568589,
%U 15861809233,51329892437,166107021401,537533612125,1739495309409,5629125066853,18216231370857,58948963008621
%N Number of connected dominating sets in the n-gear graph.
%H Colin Barker, <a href="/A287431/b287431.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-6,4).
%F From _Colin Barker_, May 25 2017: (Start)
%F G.f.: x*(7 - 7*x - 32*x^2 + 12*x^3)/((1 - x)^2*(1 - 2*x - 4*x^2)).
%F a(n) = 1 + 4*n + (1-sqrt(5))^n + (1+sqrt(5))^n.
%F a(n) = 4*a(n-1) - a(n-2) - 6*a(n-3) + 4*a(n-4) for n>4. (End)
%t LinearRecurrence[{4, -1, -6, 4}, {7, 21, 45, 129}, 20]
%t Table[2^n LucasL[n] + 4 n + 1, {n, 20}]
%o (PARI) Vec(x*(7 - 7*x - 32*x^2 + 12*x^3) / ((1 - x)^2*(1 - 2*x - 4*x^2)) + O(x^30)) \\ _Colin Barker_, May 25 2017
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, May 25 2017