

A095791


Number of digits in lazyFibonaccibinary representation of n.


10



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
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OFFSET

0,3


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 zeroone 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


LINKS

Table of n, a(n) for n=0..98.
J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965) 18.
P. Erdős and I. Joo, On the Expansion of 1 = Sum{q^(n_i)}, Period. Math. Hung. 23 (1991), no. 1, 2528.
H. H. Ferns, On the representation of integers as sums of distinct Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965) 2129.
W. Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 6069.


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) = 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 parigp program).  Benoit Cloitre, May 24 2007


EXAMPLE

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


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 *)


PROG

(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


CROSSREFS

Cf. A000045, A072649, A095791, A095792, A181632, A112309, A112310.
Sequence in context: A201052 A278044 A255121 * A238965 A036042 A162988
Adjacent sequences: A095788 A095789 A095790 * A095792 A095793 A095794


KEYWORD

nonn,base


AUTHOR

Clark Kimberling, Jun 05 2004


STATUS

approved



