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Search: id:A005207
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| A005207 |
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(F(2n-1) + F(n+1))/2 where F(n) is a Fibonacci number.
(Formerly M1183)
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+0
4
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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; listen)
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OFFSET
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1,2
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COMMENT
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.
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LINKS
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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.
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8
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FORMULA
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G.f.: (1-x)(x^2+2x-1)/((x^2+x-1)(x^2-3x+1)). a(n)=(w^(2n-1)+w^(1-2n)+w^(n+1)-(-w)^(-1-n))/(4w-2) where w=(1+sqrt(5))/2. a(n)=4a(n-1)-3a(n-2)-2a(n-3)+a(n-4).
a(n)=2/5*Sum(k, 1, 4, Sin(Pi*k/5)^2(1+2Cos(Pi*k/5))^n) - Herbert Kociemba (kociemba(AT)t-online.de), May 31 2004
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MAPLE
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A005207:=-(1-2*z-z**2+z**3)/(z**2-3*z+1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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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PROGRAM
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(PARI) a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2
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CROSSREFS
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a(-1-2n)=A027994(2n), a(-2n)=A059512(2n+1).
Sequence in context: A091600 A048285 A051529 this_sequence A094286 A094287 A094288
Adjacent sequences: A005204 A005205 A005206 this_sequence A005208 A005209 A005210
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 04 2002
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