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.
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
KEYWORD
nonn,easy
AUTHOR
Cesar Bautista, Apr 14 2012
STATUS
approved