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A028859
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a(n+2) = 2*a(n+1) + 2*a(n).
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29
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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)
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OFFSET
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0,2
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COMMENTS
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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
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
Contribution from Gary W. Adamson, Jul 28 2009: (Start)
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 - ...). (End)
Equals row 2 of the array in A180165, and the INVERTi transform of A125145. [From 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
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REFERENCES
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M. Benoumhani, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1. - From N. J. A. Sloane, Oct 09 2012
P. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, Journal of Integer Sequences, 10 (2007), Article 07.2.8.
S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 73).
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
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; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..1000
Joerg Arndt, Fxtbook, section 14.9 "Strings with no two consecutive zeros", pp.318-320.
Tanya Khovanova, Recursive Sequences
Index to sequences with linear recurrences with constant coefficients, signature (2,2).
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FORMULA
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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. [From Milan Janjic, May 08 2010]
a(n) = 3^n - A186244(n). - Toby Gottfried, Mar 07 2013
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MAPLE
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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); (Deutsch)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A180165, A125145, A026150, A030195, A080040, A083337, A106435, A108898.
Cf. A155020 (same sequence with term 1 prepended).
Sequence in context: A055887 A024581 * A155020 A014397 A048503 A200752
Adjacent sequences: A028856 A028857 A028858 * A028860 A028861 A028862
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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