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A003688 a(n) = 3*a(n-1) + a(n-2). +0
6
1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564, 21932293, 72437443, 239244622, 790171309, 2609758549, 8619446956, 28468099417, 94023745207, 310539335038, 1025641750321 (list; graph; listen)
OFFSET

1,2

COMMENT

Number of 2-factors in K_3 X P_n.

Form the graph with matrix [1,1,1,1;1,1,1,0;1,1,0,1;1,0,1,1]. The sequence 1,1,4,13... with g.f. (1-2x)/(1-3x-x^2) counts closed walks of length n at the vertex of degree 5. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

LINKS

Joerg Arndt, Fxtbook

Tanya Khovanova, Recursive Sequences

F. Faase, Counting Hamilton cycles in product graphs

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 419

FORMULA

a(n)=(1/2-sqrt(13)/26)(3/2+sqrt(13)/2)^n+(1/2+sqrt(13)/26)(3/2-sqrt(13)/2)^n - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

a(n)=Sum_{k, 0<=k<=n}2^k*A055830(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2006

Starting (1, 1, 4, 13, 43, 142, 469,...), = row sums (unsigned) of triangle A136159. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 16 2007

MAPLE

with(combinat): a:=n->fibonacci(n, 3)-2*fibonacci(n-1, 3): seq(a(n), n=2..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008

MATHEMATICA

a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}] (from Robert G. Wilson v Jan 13 2005)

CROSSREFS

Partial sums of A052906. Pairwise sums of A006190.

Cf. A136159.

Adjacent sequences: A003685 A003686 A003687 this_sequence A003689 A003690 A003691

Sequence in context: A047144 A072307 A121486 this_sequence A033434 A113986 A042767

KEYWORD

nonn

AUTHOR

Frans Faase (Frans_LiXia(AT)wxs.nl)

EXTENSIONS

Formula added Aug 15 1997 by Olivier Gerard

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Last modified October 13 02:37 EDT 2008. Contains 145008 sequences.


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