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A014217 Floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio. 32

%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 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 A000045(n)*phi . - _Benoit Cloitre_, Jun 19 2005

%C a(n) is also the number of circles curvature (rounded down) inscibed in kite arranged as spiral form, starting with a unit circle. See illustration in links. - _Kival Ngaokrajang_, Aug 29 2913

%D Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

%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>, arXiv:0709.2394

%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).

%H Kival Ngaokrajang, <a href="/A014217/a014217.pdf">Illustration for n = 0..7</a>

%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 a(n) = A000032(n)-(1+(-1)^n)/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 17 2003

%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]

%F a(n+2*k)-a(n) = A203976(k)*A000032(n+k) if k odd; a(n+2*k)-a(n) = A203976(k)*A000045(n+k) if k even; for k>0. - _Paul Curtz, Jun 05 2013

%F a(n) = A052952(n) - A052952(n-2) +A052952(n-3). - _R. J. Mathar_, Jun 13 2013

%F a(n+6) - a(n-6) = 40*A000045(n), case k=6 of my formula above. - _Paul Curtz_, Jun 13 2013

%F a(n-3) + a(n+3) = A153382(n). - _Paul Curtz_, Jun 17 2013

%F a(n-1) + a(n+2) = A022319(n). - _Paul Curtz_, Jun 17 2013

%p A014217 := proc(n)

%p option remember;

%p if n <= 3 then

%p op(n+1,[1,1,2,4]) ;

%p else

%p procname(n-1)+2*procname(n-2)-procname(n-3)-procname(n-4) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 23 2013

%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, A226328.

%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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Last modified April 16 03:40 EDT 2014. Contains 240534 sequences.