A286910 Number of independent vertex sets and vertex covers in the n-antiprism graph.
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
Offset: 0
Keywords
Links
- 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).
Programs
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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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Mathematica
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 *) -
PARI
Vec((-2*x^2 - 2*x + 3)/(-x^3 - 2*x^2 - x + 1)+O(x^30))
Formula
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
Comments