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A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Also the number of vertex covers. - Eric W. Weisstein, Jan 04 2014
LINKS
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
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
Sequence in context: A018470 A281438 A120748 * A193649 A186341 A262326
KEYWORD
nonn,easy
AUTHOR
Cesar Bautista, Apr 14 2012
STATUS
approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)