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A003688
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a(n) = 3*a(n-1) + a(n-2).
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12
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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; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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. (1-2*x)/(1-3*x-x^2) counts closed walks of length n at the vertex of degree 5. - Paul Barry, Oct 02 2004
a(n) is term (1,1) in M^n, where M is the 3x3 matrix [1,1,2; 1,1,1; 1,1,1]. [From Gary W. Adamson, Mar 12 2009]
a(n)=b(n)+b(n-1) where $\{b(n)\}$ is the sequence A006190 [From Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Nov 26 2009]
Starting with 1, INVERT transform of A003945: (1, 3, 6, 12, 24,...). [Gary W. Adamson, Aug 05 2010]
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REFERENCES
| F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
S. Falcon and A. Plaza, On the Fibonacci $k$-numbers, Chaos, Solitons \& Fractals, 32(5) (2007), 1615-1624. [From Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Nov 26 2009]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Joerg Arndt, Fxtbook
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamilton cycles in product graphs
F. Faase, Results from the counting program
F. Faase, Counting Hamilton cycles in product graphs
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 419
Tanya Khovanova, Recursive Sequences
Index to sequences with linear recurrences with constant coefficients, signature (3,1).
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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, Oct 02 2004
a(n)=Sum_{k, 0<=k<=n}2^k*A055830(n,k) . - Philippe DELEHAM, Oct 18 2006
Starting (1, 1, 4, 13, 43, 142, 469,...), row sums (unsigned) of triangle A136159. - Gary W. Adamson, Dec 16 2007
G.f.: x*(1+x)/(1-3*x-x^2). [From Philippe DELEHAM, Nov 03 2008]
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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
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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)
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PROG
| (MAGMA) [ n eq 1 select 1 else n eq 2 select 4 else 3*Self(n-1)+Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 19 2011
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CROSSREFS
| Partial sums of A052906. Pairwise sums of A006190.
Cf. A136159.
Cf. A003945 [From Gary W. Adamson, Aug 05 2010]
Sequence in context: A072307 A121486 A188176 * A033434 A113986 A149426
Adjacent sequences: A003685 A003686 A003687 * A003689 A003690 A003691
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KEYWORD
| nonn,easy
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AUTHOR
| Frans Faase (Frans_LiXia(AT)wxs.nl)
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EXTENSIONS
| Formula added Aug 15 1997 by Olivier Gerard
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