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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) inscribed in kite arranged as spiral form, starting with a unit circle. See illustration in links. - _Kival Ngaokrajang_, Aug 29 2913

%C a(n) is the n-th Lucas number (A000032) if n is odd, and a(n) is the n-th Lucas number minus 1 if n is even. (Mario Catalani's formula below expresses this fact.) This is related to the fact that the powers of phi approach the values of the Lucas numbers, the odd powers from above and the even powers from below. - _Geoffrey Caveney_, Apr 18 2014

%H T. D. Noe, <a href="/A014217/b014217.txt">Table of n, a(n) for n = 0..300</a>

%H Mohammad K. Azarian, <a href="http://www.math-cs.ucmo.edu/~mjms/1998.3/prob.ps">Problem 123</a>, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. <a href="http://www.math-cs.ucmo.edu/~mjms/2000.1/soln.ps">Solution</a> published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

%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}] (* _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 September 23 14:27 EDT 2014. Contains 247171 sequences.