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 A072649 n occurs Fibonacci(n) times (cf. A000045). 50
 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of digits in Zeckendorf-binary representation of n. E.g., the Zeckendorf representation of 12 is 8+3+1, which in binary notation is 10101, which consists of 5 digits. - Clark Kimberling, Jun 05 2004 First position where value n occurs is A000045(n+1), i.e., a(A000045(n)) = n-1, for n >= 2 and a(A000045(n)-1) = n-2, for n >= 3. This is the number of distinct Fibonacci numbers greater than 0 which are less than or equal to n. - Robert G. Wilson v, Dec 10 2006 The smallest nondecreasing sequence a(n) such that a(Fibonacci(n-1)) = n. - Tanya Khovanova, Jun 20 2007 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 Popular Computing (Calabasas, CA), A Coding Exercise (from a suggestion by R. W. Hamming), Vol. 5 (No. 54, Sep 1977), p. PC55-18. FORMULA G.f.: (Sum_{n>1} x^Fibonacci(n))/(1-x). - Michael Somos, Apr 25, 2003 a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) - 1, n >= 1, where phi is the golden ratio. Alternatively, a(n) = floor(arcsinh(sqrt(5)*n/2)/log(phi)) - 1. Also a(n) = A108852(n) - 2. - Hieronymus Fischer, May 02 2007 a(n) = -1 + floor( log_phi( (n+0.2)*sqrt(5) ) ), where log_phi(x) is the logarithm to the base (1+sqrt(5))/2. - Ralf Stephan, May 14 2007 EXAMPLE 1, 1, then F(2) 2's, then F(3) 3's, then F(4) 4's, ..., then F(k) k's, ... MAPLE a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n)        while combinat[fibonacci](j+1)<=n do od; (j-1)     end: seq(a(n), n=1..120);  # Alois P. Heinz, Mar 18 2013 MATHEMATICA Table[Table[n, {Fibonacci[n]}], {n, 10}] // Flatten (* Robert G. Wilson v, Jan 14 2007 *) PROG (PARI) a(n) = -1+floor(log(((n+0.2)*sqrt(5)))/log((1+sqrt(5))/2)) (PARI) a(n)=local(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2) (Haskell) a072649 n = a072649_list !! (n-1) a072649_list = f 1 where    f n = (replicate (fromInteger \$ a000045 n) n) ++ f (n+1) -- Reinhard Zumkeller, Jul 04 2011 (Python) from sympy import fibonacci def a(n):     if n<1: return 0     m=0     while fibonacci(m)<=n: m+=1     return m-2 print [a(n) for n in xrange(1, 101)] # Indranil Ghosh, Jun 09 2017 (MIT Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b))))))) ;; (The implementation below is better) (Scheme) (define (A072649 n) (if (<= n 3) n (let loop ((k 5)) (if (> (A000045 k) n) (- k 2) (loop (+ 1 k)))))) ;; (Use this with the memoized implementation of A000045 given under that entry. No floating point arithmetic is involved). - Antti Karttunen, Oct 06 2017 CROSSREFS Cf. A000045, A095791, A095792. Used to construct A003714. Cf. also A002024, A072643, A072648, A072650. Cf. A131234. Partial sums: A256966, A256967. Sequence in context: A220348 A274010 A213711 * A266082 A105195 A257569 Adjacent sequences:  A072646 A072647 A072648 * A072650 A072651 A072652 KEYWORD nonn,look AUTHOR Antti Karttunen, Jun 02 2002 EXTENSIONS Typo fixed by Charles R Greathouse IV, Oct 28 2009 STATUS approved

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Last modified October 19 09:18 EDT 2018. Contains 316339 sequences. (Running on oeis4.)