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 A028859 a(n+2) = 2*a(n+1) + 2*a(n); a(0) = 1, a(1) = 3. 34
 1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of words of length n without adjacent 0's from the alphabet {0,1,2}. For example, a(2) counts 01,02,10,11,12,20,21,22. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 12 2001 Individually, both this sequence and A002605 are convergents to 1+sqrt(3). Mutually, both sequences are convergents to 2+sqrt(3) and 1+sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001 [Can someone clarify what is meant by the obscure second phrase, "Mutually..."? - M. F. Hasler, Aug 06 2018] Add a loop at two vertices of the graph C_3=K_3. A028859(n) counts walks of length n+1 between these vertices. - Paul Barry, Oct 15 2004 Prefaced with a 1 as (1 + x + 3x^2 + 8x^3 + 22x^4 + ...) = 1 / (1 - x - 2x^2 - 3x^3 - 5x^4 - 8x^5 - 13x^6 - 21x^7 - ...). - Gary W. Adamson, Jul 28 2009 Equals row 2 of the array in A180165, and the INVERTi transform of A125145. - Gary W. Adamson, Aug 14 2010 Pisano period lengths: 1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, .... - R. J. Mathar, Aug 10 2012 Also the number of independent vertex sets and vertex covers in the n-centipede graph. - Eric W. Weisstein, Sep 21 2017 REFERENCES S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 73). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 Joerg Arndt, Matters Computational (The Fxtbook), section 14.9 "Strings with no two consecutive zeros", pp.318-320. C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8. Moussa Benoumhani, On the Modes of the Independence Polynomial of the Centipede, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1. Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8. David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13. Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. 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 Eric Weisstein's World of Mathematics, Centipede Graph Eric Weisstein's World of Mathematics, Independent Vertex Set Eric Weisstein's World of Mathematics, Vertex Cover Index entries for linear recurrences with constant coefficients, signature (2,2). FORMULA a(n) = a(n-1) + A052945(n) = A002605(n) + A002605(n-1); generating function = -(x+1)/(2*x^2+2*x-1). a(n)=[(1+sqrt(3))^(n+2)-(1-sqrt(3))^(n+2)]/(4*sqrt(3)). - Emeric Deutsch, Feb 01 2005 If p[i]=fibonacci(i+1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010 a(n) = 3^n - A186244(n). - Toby Gottfried, Mar 07 2013 MAPLE a[0]:=1:a[1]:=3:for n from 2 to 24 do a[n]:=2*a[n-1]+2*a[n-2] od: seq(a[n], n=0..24); # Emeric Deutsch MATHEMATICA a[n_]:=(MatrixPower[{{1, 3}, {1, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *) Table[2^(n - 1) Hypergeometric2F1[(1 - n)/2, -n/2, -n, -2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *) LinearRecurrence[{2, 2}, {1, 3}, 20] (* Eric W. Weisstein, Jun 14 2017 *) PROG (Haskell) a028859 n = a028859_list !! n a028859_list =    1 : 3 : map (* 2) (zipWith (+) a028859_list (tail a028859_list)) -- Reinhard Zumkeller, Oct 15 2011 (PARI) a(n)=([1, 3; 1, 1]^n*[2; 1])[2, 1] \\ Charles R Greathouse IV, Mar 27 2012 (PARI) A028859(n)=([1, 1]*[2, 2; 1, 0]^n)[1] \\ M. F. Hasler, Aug 06 2018 CROSSREFS Cf. A180165, A125145, A026150, A030195, A080040, A083337, A106435, A108898. Cf. A155020 (same sequence with term 1 prepended). Cf. A002605. Sequence in context: A278612 A024581 * A155020 A255705 A014397 A048503 Adjacent sequences:  A028856 A028857 A028858 * A028860 A028861 A028862 KEYWORD nonn,easy,changed AUTHOR EXTENSIONS Definition completed by M. F. Hasler, Aug 06 2018 STATUS approved

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Last modified August 19 11:58 EDT 2018. Contains 313862 sequences. (Running on oeis4.)