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A286910
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Number of independent vertex sets and vertex covers in the n-antiprism graph.
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2
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3, 1, 5, 10, 21, 46, 98, 211, 453, 973, 2090, 4489, 9642, 20710, 44483, 95545, 205221, 440794, 946781, 2033590, 4367946, 9381907, 20151389, 43283149, 92967834, 199685521, 428904338, 921243214, 1978737411, 4250128177, 9128846213, 19607839978, 42115660581
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OFFSET
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0,1
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COMMENTS
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Sequence extrapolated to n=0 using recurrence.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..500
Haoliang Wang, Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (1,2,1).
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FORMULA
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a(n) = a(n-1) + 2*a(n-2) + a(n-3) for n>=3.
G.f.: (2*x^2 + 2*x - 3)/(x^3 + 2*x^2 + x - 1).
a(n) = n*Sum_{k=1..n} C(2*k,n-k)/k, a(0)=3. - Vladimir Kruchinin, Jun 13 2020
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MATHEMATICA
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CoefficientList[Series[(- 2 x^2 - 2 x + 3) / (- x^3 - 2 x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, May 16 2017 *)
LinearRecurrence[{1, 2, 1}, {3, 1, 5}, 40] (* Vincenzo Librandi, May 16 2017 *)
Table[RootSum[-1 - 2 # - #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, Aug 16 2017 *)
RootSum[-1 - 2 # - #^2 + #^3 &, #^Range[20] &] (* Eric W. Weisstein, Aug 16 2017 *)
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PROG
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(PARI)
Vec((-2*x^2 - 2*x + 3)/(-x^3 - 2*x^2 - x + 1)+O(x^30))
(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
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CROSSREFS
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Cf. A051927, A182143, A192742, A284700.
Sequence in context: A065229 A233037 A275999 * A093905 A324017 A063853
Adjacent sequences: A286907 A286908 A286909 * A286911 A286912 A286913
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KEYWORD
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nonn
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AUTHOR
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Andrew Howroyd, May 15 2017
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STATUS
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approved
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