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A014217 a(n) = floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio. 35
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, 54018521, 87403802 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Floor{lim k->oo {Fibonacci(k)/Fibonacci(k-n)}}. - Jon Perry, Jun 10 2003

For n>1 a(n) is the maximum element in the continued fraction for A000045(n)*phi. - Benoit Cloitre, Jun 19 2005

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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..4784 (first 301 terms from T. D. Noe)

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.

Ayman A. El-Okaby, Exceptional Lie Groups, E-infinity Theory and Higgs Boson, arXiv:0709.2394 [physics.gen-ph], 2007.

G. Harman, One hundred years of normal numbers

Kival Ngaokrajang, Illustration for n = 0..7

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

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 = a(n-1) + a(n-2) + A000035(n).

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

G.f.: (1-x^2+x^3)/((1+x)(1-x)(1-x-x^2)). - R. J. Mathar, Sep 06 2008

a(2n-1) = (Fibonacci(4n+1)-2)/Fibonacci(2n+2). - Gary Detlefs, Feb 16 2011

a(n) = floor(Fibonacci(2n+3)/Fibonacci(n+3)). - Gary Detlefs, Feb 28 2011

a(2n) = Fibonacci(2*n-1)+Fibonacci(2*n+1)-1. - Gary Detlefs, Mar 10 2011

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

a(n) = A052952(n) - A052952(n-2) + A052952(n-3). - R. J. Mathar, Jun 13 2013

a(n+6) - a(n-6) = 40*A000045(n), case k=6 of my formula above. - Paul Curtz, Jun 13 2013

a(n-3) + a(n+3) = A153382(n). - Paul Curtz, Jun 17 2013

a(n-1) + a(n+2) = A022319(n). - Paul Curtz, Jun 17 2013

For k>0, a(2k) = A169985(2k)-1 and a(2k+1) = A169985(2k+1) (which is equivalent to Catalani's 2003 formula). - Danny Rorabaugh, Apr 15 2015

a(n) = ((-1)^(1+n)-1)/2 + ((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n. - Colin Barker, Nov 05 2017

MAPLE

A014217 := proc(n)

    option remember;

    if n <= 3 then

        op(n+1, [1, 1, 2, 4]) ;

    else

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

    end if;

end proc: # R. J. Mathar, Jun 23 2013

#

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-1|2|1>>^n. <<1, 1, 2, 4>>)[1, 1]:

seq(a(n), n=0..40);  # Alois P. Heinz, Oct 12 2017

MATHEMATICA

Table[Floor[GoldenRatio^n], {n, 0, 36}] (* Vladimir Joseph Stephan Orlovsky, Dec 12 2008 *)

LinearRecurrence[{1, 2, -1, -1}, {1, 1, 2, 4}, 40] (* Jean-Fran├žois Alcover, Nov 05 2017 *)

PROG

(PARI) for (n=0, 20, print1(fibonacci(1000)\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

(Sage) [floor(golden_ratio^n) for n in range(37)] # Danny Rorabaugh, Apr 19 2015

CROSSREFS

Cf. A000045, A020956, A052952, A057146, A062114, A169985, A169986, A226328.

Sequence in context: A018144 A115315 A004698 * A034297 A026636 A026658

Adjacent sequences:  A014214 A014215 A014216 * A014218 A014219 A014220

KEYWORD

nonn,easy,nice,changed

AUTHOR

Clark Kimberling

EXTENSIONS

Corrected by T. D. Noe, Nov 09 2006

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

STATUS

approved

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Last modified November 18 04:49 EST 2017. Contains 294852 sequences.