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%I #26 Sep 08 2022 08:46:19
%S 3,1,5,10,21,46,98,211,453,973,2090,4489,9642,20710,44483,95545,
%T 205221,440794,946781,2033590,4367946,9381907,20151389,43283149,
%U 92967834,199685521,428904338,921243214,1978737411,4250128177,9128846213,19607839978,42115660581
%N Number of independent vertex sets and vertex covers in the n-antiprism graph.
%C Sequence extrapolated to n=0 using recurrence.
%H Andrew Howroyd, <a href="/A286910/b286910.txt">Table of n, a(n) for n = 0..500</a>
%H Haoliang Wang, Robert Simon, <a href="https://doi.org/10.1145/3267129.3267134">The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems</a>, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.
%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/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1).
%F a(n) = a(n-1) + 2*a(n-2) + a(n-3) for n>=3.
%F G.f.: (2*x^2 + 2*x - 3)/(x^3 + 2*x^2 + x - 1).
%F a(n) = n*Sum_{k=1..n} C(2*k,n-k)/k, a(0)=3. - _Vladimir Kruchinin_, Jun 13 2020
%t CoefficientList[Series[(- 2 x^2 - 2 x + 3) / (- x^3 - 2 x^2 - x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 16 2017 *)
%t LinearRecurrence[{1, 2, 1}, {3, 1, 5}, 40] (* _Vincenzo Librandi_, May 16 2017 *)
%t Table[RootSum[-1 - 2 # - #^2 + #^3 &, #^n &], {n, 20}] (* _Eric W. Weisstein_, Aug 16 2017 *)
%t RootSum[-1 - 2 # - #^2 + #^3 &, #^Range[20] &] (* _Eric W. Weisstein_, Aug 16 2017 *)
%o (PARI)
%o Vec((-2*x^2 - 2*x + 3)/(-x^3 - 2*x^2 - x + 1)+O(x^30))
%o (Magma) I:=[3,1,5]; [n le 3 select I[n] else Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..33]]; // _Vincenzo Librandi_, May 16 2017
%Y Cf. A051927, A182143, A192742, A284700.
%K nonn
%O 0,1
%A _Andrew Howroyd_, May 15 2017
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