%I #55 Sep 08 2022 08:45:54

%S 1,3,5,15,33,83,197,479,1153,2787,6725,16239,39201,94643,228485,

%T 551615,1331713,3215043,7761797,18738639,45239073,109216787,263672645,

%U 636562079,1536796801,3710155683,8957108165,21624372015,52205852193,126036076403,304278004997

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

%C Also the number of vertex covers. - _Eric W. Weisstein_, Jan 04 2014

%H Cesar Bautista, <a href="/A182143/b182143.txt">Table of n, a(n) for n = 0..1000</a>

%H C. Bautista-Ramos and C. Guillen-Galvan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Bautista/bautista4.html">Fibonacci numbers of generalized Zykov sums</a>, J. Integer Seq., 15 (2012), Article 12.7.8

%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/MoebiusLadder.html">Moebius Ladder</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 (1,3,1).

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

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

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

%F From _Peter Bala_, Jun 29 2015: (Start)

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

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

%F 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)

%t Table[(1 + Sqrt[2])^n + (1 - Sqrt[2])^n - (-1)^n, {n, 0, 30}] (* _Bruno Berselli_, Apr 14 2012 *)

%t Table[LucasL[n, 2] - (-1)^n, {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 15 2016 *)

%t LinearRecurrence[{1, 3, 1}, {1, 3, 5}, 20] (* _Eric W. Weisstein_, Mar 31 2017 *)

%t CoefficientList[Series[(-1 - 2 x + x^2)/(-1 + x + 3 x^2 + x^3), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)

%o (PARI) Vec((x^2-2*x-1)/((x+1)*(x^2+2*x-1))+O(x^31)) \\ _Bruno Berselli_, Apr 14 2012

%o (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

%Y Cf. A000129, A001333, A002203, A098600.

%K nonn,easy

%O 0,2

%A _Cesar Bautista_, Apr 14 2012