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A072649 n occurs Fibonacci(n) times (cf. A000045). 36
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)

(MIT Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b)))))))

(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

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 March 29 06:34 EDT 2017. Contains 284250 sequences.