A095792 -id:A095792 - cp's OEIS frontend

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

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

Offset: 1

Antti Karttunen, Jun 02 2002

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

			1, 1, then F(2) 2's, then F(3) 3's, then F(4) 4's, ..., then F(k) k's, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 4.
Popular Computing (Calabasas, CA), A Coding Exercise (from a suggestion by R. W. Hamming), Vol. 5 (No. 54, Sep 1977), p. PC55-18.

Cf. A000045, A095791, A095792.
Cf. A001622 (golden ratio phi), A073010.
Used to construct A003714. Cf. also A002024, A072643, A072648, A072650.
Cf. A131234.
Partial sums: A256966, A256967.

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

A072649 := proc(n)
    local j;
    for j from ilog[(1+sqrt(5))/2](n) while combinat[fibonacci](j+1)<=n do
    end do;
    j-1
end proc:
seq(A072649(n), n=1..120);  # Alois P. Heinz, Mar 18 2013

Table[Table[n, {Fibonacci[n]}], {n, 10}] // Flatten (* Robert G. Wilson v, Jan 14 2007 *)
a[n_] := Module[{j}, For[j = Floor@Log[GoldenRatio, n], Fibonacci[j+1] <= n, j++]; j-1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 17 2022, after Alois P. Heinz *)

a(n) = -1+floor(log(((n+0.2)*sqrt(5)))/log((1+sqrt(5))/2))

a(n)=local(m); if(n<1,0,m=0; until(fibonacci(m)>n,m++); m-2)

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 range(1, 101)]) # Indranil Ghosh, Jun 09 2017

def A072649(n):
    a, b, c = 0, 1, -2
    while a <= n:
        a, b = b, a+b
        c += 1
    return c # Chai Wah Wu, Nov 04 2024
(MIT/GNU Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b))))))) ;; (The implementation below is better)

(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

G.f.: (Sum_{n>1} x^Fibonacci(n))/(1-x). - Michael Somos, Apr 25 2003
From Hieronymus Fischer, May 02 2007: (Start)
a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) - 1, where phi is A001622.
a(n) = floor(arcsinh(sqrt(5)*n/2)/log(phi)) - 1.
a(n) = A108852(n) - 2. (End)
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
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 18 2024

Typo fixed by Charles R Greathouse IV, Oct 28 2009

A095791 Number of digits in lazy-Fibonacci-binary representation of n (A104326).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 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, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Clark Kimberling, Jun 05 2004

The lazy Fibonacci representation of n >= 0 is obtained by replacing every string of 0's in the binary representation of n by a single 0, thus obtaining a finite zero-one sequence (d(2), d(3), d(4), ..., d(k)), and then forming d(2)*F(2) + d(3)*F(3) + ... + d(k)*F(k), as in the Mathematica program. The lazy Fibonacci representation is often called the maximal Fibonacci representation, in contrast to the Zeckendorf representation, also called the minimal Fibonacci representation. - Clark Kimberling, Mar 04 2015

Regarding the References, the lazy Fibonacci representation is sometimes attributed to Erdős and Joo, but it is also found in Brown and Ferns. - Clark Kimberling, Mar 04 2015

			The lazy Fibonacci representation of 14 is 8+3+2+1, which in binary notation is 10111, which consists of 5 digits.

Amiram Eldar, Table of n, a(n) for n = 0..10000
J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965), 1-8.
P. Erdős and I. Joo, On the Expansion of 1 = Sum{q^(-n_i)}, Period. Math. Hung. 23 (1991), no. 1, 25-28.
H. H. Ferns, On the representation of integers as sums of distinct Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965), 21-29.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik, Vol. 78, No. 1 (2021), pp. 1-8.
Wolfgang Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 60-69.

Cf. A000045, A070939, A072649, A095792, A104326, A112309, A112310, A181632.

t=DeleteCases[IntegerDigits[-1+Range[200],2],{_,0,0,_}];
A181632=Flatten[t]
A095791=Map[Length,t]
A112309=Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]]+1],0]&,t]
A112310=Map[Length,A112309]
(* Peter J. C. Moses, Mar 03 2015 *)

a(n)=if(n<2,1,a(floor(n*(-1+sqrt(5))/2))+1) \\ Benoit Cloitre, Dec 17 2006

a(n)=if(n<0,0,c=1;s=n;while(floor(s*2/(1+sqrt(5)))>0,c++;s=floor(s*2/(1+sqrt(5))));c) \\ Benoit Cloitre, May 24 2007

1, 1, then F(3) 2's, then F(4) 3's, then F(5) 4's, ..., then F(k+1) k's, ...
a(0)=a(1)=1 then a(n) = a(floor(n/tau))+1 where tau=(1+sqrt(5))/2. - Benoit Cloitre, Dec 17 2006
a(n) is the least k such that f^(k)(n)=0 where f^(k+1)(x)=f(f^(k)(x)) and f(x)=floor(x/phi) where phi=(1+sqrt(5))/2 (see PARI/GP program). - Benoit Cloitre, May 24 2007
a(n) = A070939(A104326(n)). - Amiram Eldar, Oct 10 2023

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

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A095791 Number of digits in lazy-Fibonacci-binary representation of n (A104326).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula