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A005207
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(F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.
(Formerly M1183)
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5
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1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933, 1763581, 4615823, 12082291, 31628466, 82798926, 216761547, 567474769, 1485645049, 3889431721, 10182603746, 26658304492, 69792188337
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of block fountains with exactly n coins in the base when mirror image fountains are identified. [From Michael Woltermann (mwoltermann(AT)washjeff.edu), Oct 06 2010]
a(n) = C(F(n+1)+1,2) + C(F(n)+1,2) = pairwise sums of A033192. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 06 2003
Number of (3412,54312)- and (3412,45321)-avoiding involutions in S_{n+1}. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 06 2003
Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 1. - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
Contribution from Paul Barry (pbarry(AT)wit.ie), Dec 17 2009: (Start)
The sequence 1,1,2,4,9,... has g.f. 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x))))=(1-3*x+x^2+x^2)/(1-4*x+3*x^2+2*x^3-x^4), and general term (A001519(n)+A000045(n+1))/2. It is the binomial trasnform of A001519 aerated. (End)
The Kn3 and Kn4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence. [Johannes W. Meijer, Jul 14 2011]
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REFERENCES
| M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.
Problem 1826, Mathematics Magazine, 83 (2010), 304-305. [From Michael Woltermann (mwoltermann(AT)washjeff.edu), Oct 06 2010]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..300
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, Arxiv preprint arXiv:1105.3713, 2011.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (4,-3,-2,1)
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FORMULA
| G.f.: -x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1)).
a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3)+a(n-4).
a(n) = (w^(2*n-1)+w^(1-2*n)+w^(n+1)-(-w)^(-1-n))/(4*w-2) where w=(1+sqrt(5))/2.
a(n) = 2/5*sum(k=1..4, sin(Pi*k/5)^2*(1+2*cos(Pi*k/5))^n ) - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
a(-1-2*n) = A027994(2*n); a(-2*n)=A059512(2*n+1).
Let M = an infinite tridiagonal matrix with all 1's in the super and main diagonals and [1,1,1,0,0,0,...] in the sub diagonal. Let V = vector [1,0,0,0,...]. The sequence is generated as leftmost column of M*V iterates. [From Gary W. Adamson, (qntmpkt(AT)yahoo.com) Jun 07 2011]
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MAPLE
| A005207:=-(1-2*z-z**2+z**3)/(z**2-3*z+1)/(z**2+z-1); [S. Plouffe in his 1992 dissertation with offset 0.]
a:= n-> (Matrix([[1, 1, 1, 3]]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -3, -2, 1][i] else 0 fi)^n)[1, 2]: seq (a(n), n=1..28); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 06 2008]
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PROG
| (PARI) a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2
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CROSSREFS
| Sequence in context: A176334 A048285 A051529 * A094286 A094287 A094288
Adjacent sequences: A005204 A005205 A005206 * A005208 A005209 A005210
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 04 2002
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