A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0).

1, 3, 5, 15, 33, 83, 197, 479, 1153, 2787, 6725, 16239, 39201, 94643, 228485, 551615, 1331713, 3215043, 7761797, 18738639, 45239073, 109216787, 263672645, 636562079, 1536796801, 3710155683, 8957108165, 21624372015, 52205852193, 126036076403, 304278004997

Offset

0,2

Comments

Also the number of vertex covers. - Eric W. Weisstein, Jan 04 2014

Links

Cesar Bautista, Table of n, a(n) for n = 0..1000

C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8

Eric Weisstein's World of Mathematics, Independent Vertex Set

Eric Weisstein's World of Mathematics, Moebius Ladder

Eric Weisstein's World of Mathematics, Vertex Cover

Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Formula

G.f.: (x^2-2*x-1)/((x+1)*(x^2+2*x-1)).

a(n) = (1+sqrt(2))^n + (1-sqrt(2))^n - (-1)^n = A002203(n) - (-1)^n.

a(n) = a(n-1) + 3*a(n-2) + a(n-3) with a(0)=1, a(1)=3, a(2)=5.

From Peter Bala, Jun 29 2015: (Start)

a(n) = Pell(n-1) + Pell(n+1) - (-1)^n.

a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 8*x + 8*x^2))/2 )^n.

exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + ... = Sum_{n >= 0} A001333*x^n. Cf. A098600. (End)

Mathematica

Table[(1 + Sqrt[2])^n + (1 - Sqrt[2])^n - (-1)^n, {n, 0, 30}] (* Bruno Berselli, Apr 14 2012 *)
Table[LucasL[n, 2] - (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
LinearRecurrence[{1, 3, 1}, {1, 3, 5}, 20] (* Eric W. Weisstein, Mar 31 2017 *)
CoefficientList[Series[(-1 - 2 x + x^2)/(-1 + x + 3 x^2 + x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)

Prog

(PARI) Vec((x^2-2*x-1)/((x+1)*(x^2+2*x-1))+O(x^31)) \\ Bruno Berselli, Apr 14 2012
(Magma) I:=[1, 3, 5]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2)+Self(n-3): n in [1..31]]; // Bruno Berselli, Apr 14 2012

Crossrefs

Cf. A000129, A001333, A002203, A098600.

Sequence in context: A018470 A281438 A120748 * A193649 A186341 A262326

Adjacent sequences: A182140 A182141 A182142 * A182144 A182145 A182146

Keyword

nonn,easy

Author

Cesar Bautista, Apr 14 2012

Status

approved