This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003688 a(n) = 3*a(n-1) + a(n-2), with a(1)=1 and a(2)=4. 22
 1, 4, 13, 43, 142, 469, 1549, 5116, 16897, 55807, 184318, 608761, 2010601, 6640564, 21932293, 72437443, 239244622, 790171309, 2609758549, 8619446956, 28468099417, 94023745207, 310539335038, 1025641750321, 3387464586001, 11188035508324, 36951571110973 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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]. - Gary W. Adamson, Mar 12 2009 Starting with 1, INVERT transform of A003945: (1, 3, 6, 12, 24,...). - Gary W. Adamson, Aug 05 2010 Row sums of triangle m/k.|..0.....1.....2.....3.....4.....5.....6.....7 ================================================== .0..|..1 .1..|..1.....3 .2..|..1.....3.....9 .3..|..1.....6.....9.....27 .4..|..1.....6....27.....27...81 .5..|..1.....9....27....108...81...243 .6..|..1.....9....54....108..405...243...729 .7..|..1....12....54....270..405..1458...729..2187 which is triangle for numbers 3^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012 Pisano period lengths:  1, 3, 1, 6, 12, 3, 16, 12, 6, 12, 8, 6, 52, 48, 12, 24, 16, 6, 40, 12,... - R. J. Mathar, Aug 10 2012 a(n-1) is the number of length-n strings of 4 letters {0,1,2,3} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). - Joerg Arndt, Oct 11 2012 REFERENCES F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154. LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Joerg Arndt, Matters Computational (The Fxtbook) Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018. C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8. 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, Results from the counting program Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 419 M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (3,1). 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..n} 2^k*A055830(n,k). - Philippe Deléham, 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). - Philippe Deléham, Nov 03 2008 a(n) = A006190(n) + A006190(n-1). - Sergio Falcon, Nov 26 2009 For n>=2, a(n) = F_n(3)+F_(n+1)(3), where F_n(x) is Fibonacci polynomial (cf. A049310): F_n(x)=sum{i=0,...,floor((n-1)/2)} C(n-i-1,i) x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012 G.f.: G(0)*(1+x)/(2-3*x), where G(k)= 1 + 1/(1 - (x*(13*k-9))/( x*(13*k+4) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013 a(n)^2 is the denominator of continued fraction [3,3,...,3, 5, 3,3,...3], which has n-1 3's before, and n-1 3's after, the middle 5. - Greg Dresden, Sep 18 2019 EXAMPLE G.f. = x + 4*x^2 + 13*x^3 + 43*x^4 + 142*x^5 + 469*x^6 + 1549*x^7 + 5116*x^8 + ... MAPLE with(combinat): a:=n->fibonacci(n, 3)-2*fibonacci(n-1, 3): seq(a(n), n=2..25); # Zerinvary Lajos, Apr 04 2008 MATHEMATICA a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}] (* Robert G. Wilson v, Jan 13 2005 *) LinearRecurrence[{3, 1}, {1, 4}, 30] (* Harvey P. Dale, Mar 15 2015 *) 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 (PARI) a(n)=([0, 1; 1, 3]^(n-1)*[1; 4])[1, 1] \\ Charles R Greathouse IV, Aug 14 2017 CROSSREFS Partial sums of A052906. Pairwise sums of A006190. Cf. A136159, A290948, A003945. Sequence in context: A266494 A121486 A188176 * A033434 A297928 A113986 Adjacent sequences:  A003685 A003686 A003687 * A003689 A003690 A003691 KEYWORD nonn,easy AUTHOR EXTENSIONS Formula added by Olivier Gérard, Aug 15 1997 Name clarified by Michel Marcus, Oct 16 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 20 15:29 EDT 2019. Contains 328267 sequences. (Running on oeis4.)