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

0,1

Comments

Sequence extrapolated to n=0 using recurrence.

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A051927, A182143, A192742, A284700.

Sequence in context: A065229 A233037 A275999 * A093905 A324017 A063853

Adjacent sequences: A286907 A286908 A286909 * A286911 A286912 A286913

Keyword

nonn

Author

Andrew Howroyd, May 15 2017

Status

approved