%I A014217
%S A014217 1,1,2,4,6,11,17,29,46,76,122,199,321,521,842,1364,2206,3571,5777,9349,
%T A014217 15126,24476,39602,64079,103681,167761,271442,439204,710646,1149851,
%U A014217 1860497,3010349,4870846,7881196,12752042,20633239,33385281
%N A014217 Floor( ((1+sqrt(5))/2)^n ).
%C A014217 a(n)=L(n)-(1+(-1)^n)/2, where L(n) = Lucas numbers. - Mario Catalani 
               (mario.catalani(AT)unito.it), Jan 17 2003
%C A014217 Floor{lim k->oo {Fibonacci(k)/Fibonacci(k-n)}} - Jon Perry (perry(AT)globalnet.co.uk), 
               Jun 10 2003
%C A014217 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
%C A014217 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
%C A014217 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]
%D A014217 Ayman A. El-Okaby, http://arxiv.org/pdf/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
%H A014217 T. D. Noe, <a href="b014217.txt">Table of n, a(n) for n=0..300</a>
%H A014217 G. Harman, <a href="http://at.yorku.ca/cgi-bin/amca/cadx-39">One hundred 
               years of normal numbers</a>
%F A014217 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.
%F A014217 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]
%t A014217 a[n_]:=Floor[GoldenRatio^n]; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Dec 12 2008]
%o A014217 (PARI) for (n=0,20,print1(fibonacci(1000)/(1.0*fibonacci(1000-n))","))
%Y A014217 Cf. A057146, A062114, A052952, A000045, A020956.
%Y A014217 Sequence in context: A018144 A115315 A004698 this_sequence A034297 A026636 
               A026658
%Y A014217 Adjacent sequences: A014214 A014215 A014216 this_sequence A014218 A014219 
               A014220
%K A014217 nonn,easy,nice
%O A014217 0,3
%A A014217 Clark Kimberling (ck6(AT)evansville.edu)
%E A014217 Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 09 2006
%E A014217 Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 29 2008 at 
               the suggestion of R. J. Mathar