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%I #23 Sep 08 2022 08:46:21
%S 3,11,54,179,648,2414,8809,32195,117945,431696,1579955,5783294,
%T 21168592,77482521,283608249,1038086883,3799689944,13907938601,
%U 50906985592,186333942984,682034858839,2496440225499,9137676323347,33446476209566,122423549667123
%N Number of total dominating sets in the n-antiprism graph.
%C Sequence extrapolated to n=1 using recurrence. - _Andrew Howroyd_, Apr 14 2018
%H Andrew Howroyd, <a href="/A302760/b302760.txt">Table of n, a(n) for n = 1..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,6,-3,0,0,1).
%F From _Andrew Howroyd_, Apr 14 2018: (Start)
%F a(n) = 3*a(n-1) + a(n-2) + 6*a(n-3) - 3*a(n-4) + a(n-7) for n > 7.
%F G.f.: x*(3 + 2*x + 18*x^2 - 12*x^3 + 7*x^6)/(1 - 3*x - x^2 - 6*x^3 + 3*x^4 - x^7).
%F (End)
%t CoefficientList[Series[(3 + 2 x + 18 x^2 - 12 x^3 + 7 x^6)/(1 - 3 x - x^2 - 6 x^3 + 3 x^4 - x^7), {x, 0, 24}], x] (* _Michael De Vlieger_, Apr 14 2018 *)
%t LinearRecurrence[{3, 1, 6, -3, 0, 0, 1}, {3, 11, 54, 179, 648, 2414, 8809}, 20] (* _Vincenzo Librandi_, Apr 15 2018 *)
%t Table[RootSum[-1 + 3 #^3 - 6 #^4 - #^5 - 3 #^6 + #^7 &, #^n &], {n, 30}] (* _Eric W. Weisstein_, Apr 16 2018 *)
%t RootSum[-1 + 3 #^3 - 6 #^4 - #^5 - 3 #^6 + #^7 &, #^Range[30] &] (* _Eric W. Weisstein_, Apr 16 2018 *)
%o (PARI) Vec((3 + 2*x + 18*x^2 - 12*x^3 + 7*x^6)/(1 - 3*x - x^2 - 6*x^3 + 3*x^4 - x^7) + O(x^25)) \\ _Andrew Howroyd_, Apr 14 2018
%o (Magma) I:=[3,11,54,179,648,2414,8809]; [n le 7 select I[n] else 3*Self(n-1)+Self(n-2)+6*Self(n-3)-3*Self(n-4)+Self(n-7): n in [1..30]]; // _Vincenzo Librandi_, Apr 15 2018
%Y Cf. A302255, A302652, A302763.
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, Apr 12 2018
%E a(1)-a(2) and terms a(11) and beyond from _Andrew Howroyd_, Apr 14 2018