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A014217
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Floor( ((1+sqrt(5))/2)^n ).
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32
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1, 1, 2, 4, 6, 11, 17, 29, 46, 76, 122, 199, 321, 521, 842, 1364, 2206, 3571, 5777, 9349, 15126, 24476, 39602, 64079, 103681, 167761, 271442, 439204, 710646, 1149851, 1860497, 3010349, 4870846, 7881196, 12752042, 20633239, 33385281
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=L(n)-(1+(-1)^n)/2, where L(n) = Lucas numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jan 17 2003
Floor{lim k->oo {Fibonacci(k)/Fibonacci(k-n)}} - Jon Perry (perry(AT)globalnet.co.uk), Jun 10 2003
For n>1 a(n) is the maximum element in the continued fraction for F(n)*Phi where F=A000045 and Phi=(1+sqrt(5))/2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 19 2005
An integer version of M.S. El Naschie's infinite-dimensional Markov process: d(n) = (1/d(0))^(n - 1); d(0)=(Sqrt[5] - 1)/2. - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 08 2008
From 2: successive three evens and three odds. Recurrence a(n)=a(n-1)+2a(n-2)-a(n-3)-a(n-4) also valuable for successive differences ( like for instance a(n)=3a(n-1)-3a(n-2)+2a(n-3) ). See A062724 (2, 2, 3, 5) and A098600 (1, 2, 2). [From Paul Curtz (bpcrtz(AT)free.fr), Sep 20 2008]
a(n+1) = a(n) + a(n-1) + A000035(n+1). [Reinhard Zumkeller, Jan 06 2012]
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REFERENCES
| Ayman A. El-Okaby, http://arxiv.org/abs/0709.2394, Exceptional Lie Groups, E-infinity Theory and Higgs Boson. - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 08 2008
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..300
G. Harman, One hundred years of normal numbers
Index to sequences with linear recurrences with constant coefficients, signature (1,2,-1,-1).
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FORMULA
| a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). a(n) = a(n-1) + a(n-2) + (1-(-1)^n)/2.
G.f.: (1-x^2+x^3)/((1+x)(1-x)(1-x-x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 06 2008]
a(2n-1) = (Fibonacci(4n+1)-2)/Fibonacci(2n+2). [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 16 2011]
a(n)=floor(Fibonacci(2n+3)/Fibonacci(n+3)). [From Gary Detlefs (gdetlefs(AT)aol.com), Feb 28 2011]
a(2n)=Fibonacci(2*n-1)+Fibonacci(2*n+1)-1.[From Gary Detlefs (gdetlefs(AT)aol.com), Mar 10 2011]
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MATHEMATICA
| Table[Floor[GoldenRatio^n], {n, 0, 36}] (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 12 2008 *)
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PROG
| (PARI) for (n=0, 20, print1(fibonacci(1000)/(1.0*fibonacci(1000-n))", "))
(MAGMA) [Floor( ((1+Sqrt(5))/2)^n ): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011
(Haskell)
a014217 n = a014217_list !! n
a014217_list = 1 : 1 : zipWith (+)
a000035_list (zipWith (+) a014217_list $ tail a014217_list)
-- Reinhard Zumkeller, Jan 06 2012
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CROSSREFS
| Cf. A057146, A062114, A052952, A000045, A020956, A169985, A169986.
Sequence in context: A018144 A115315 A004698 * A034297 A026636 A026658
Adjacent sequences: A014214 A014215 A014216 * A014218 A014219 A014220
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 09 2006
Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 29 2008 at the suggestion of R. J. Mathar
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