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

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

Author

Clark Kimberling, Jun 05 2004

Keywords

• nonn
• base
• easy

Comments

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

Examples

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

PARI

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

Formula

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