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%I
%S 1,1,2,4,6,11,17,29,46,76,122,199,321,521,842,1364,2206,3571,5777,
%T 9349,15126,24476,39602,64079,103681,167761,271442,439204,710646,
%U 1149851,1860497,3010349,4870846,7881196,12752042,20633239,33385281
%N Floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio.
%C a(n)=L(n)-(1+(-1)^n)/2, where L(n) = Lucas numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jan 17 2003
%C Floor{lim k->oo {Fibonacci(k)/Fibonacci(k-n)}} - _Jon Perry_, Jun 10 2003
%C For n>1 a(n) is the maximum element in the continued fraction for F(n)*phi where F=A000045. - _Benoit Cloitre_, Jun 19 2005
%H T. D. Noe, <a href="/A014217/b014217.txt">Table of n, a(n) for n = 0..300</a>
%H G. Harman, <a href="http://at.yorku.ca/cgi-bin/amca/cadx-39">One hundred years of normal numbers</a>
%H Ayman A. El-Okaby, <a href="http://arxiv.org/abs/0709.2394"> Exceptional Lie Groups, E-infinity Theory and Higgs Boson</a>
%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
%F a(n) = a(n-1) + a(n-2) + (1-(-1)^n)/2 = a(n-1) + a(n-2) + A000035(n).
%F G.f.: (1-x^2+x^3)/((1+x)(1-x)(1-x-x^2)). [From _R. J. Mathar_, Sep 06 2008]
%F a(2n-1) = (Fibonacci(4n+1)-2)/Fibonacci(2n+2). [From _Gary Detlefs_, Feb 16 2011]
%F a(n)=floor(Fibonacci(2n+3)/Fibonacci(n+3)). [From _Gary Detlefs_, Feb 28 2011]
%F a(2n)=Fibonacci(2*n-1)+Fibonacci(2*n+1)-1.[From _Gary Detlefs_, Mar 10 2011]
%t Table[Floor[GoldenRatio^n], {n, 0, 36}] (* From _Vladimir Joseph Stephan Orlovsky_, Dec 12 2008 *)
%o (PARI) for (n=0,20,print1(fibonacci(1000)/(1.0*fibonacci(1000-n))","))
%o (MAGMA) [Floor( ((1+Sqrt(5))/2)^n ): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011
%o (Haskell)
%o a014217 n = a014217_list !! n
%o a014217_list = 1 : 1 : zipWith (+)
%o a000035_list (zipWith (+) a014217_list $ tail a014217_list)
%o -- _Reinhard Zumkeller_, Jan 06 2012
%Y Cf. A057146, A062114, A052952, A000045, A020956, A169985, A169986.
%K nonn,easy,nice
%O 0,3
%A _Clark Kimberling_
%E Corrected by _T. D. Noe_, Nov 09 2006
%E Edited by _N. J. A. Sloane_, Aug 29 2008 at the suggestion of R. J. Mathar
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