

A104326


Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00.


38



0, 1, 10, 11, 101, 110, 111, 1010, 1011, 1101, 1110, 1111, 10101, 10110, 10111, 11010, 11011, 11101, 11110, 11111, 101010, 101011, 101101, 101110, 101111, 110101, 110110, 110111, 111010, 111011, 111101, 111110, 111111, 1010101
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OFFSET

0,3


COMMENTS

Whereas the Zeckendorf (binary) rep (A014417) has no consecutive 1's (no two consecutive Fibonacci numbers in a set whose sum is n), the Dual Zeckendorf Representation has no consecutive 0's. Also called the Maximal (Binary) Fibonacci Representation, the Zeckendorf rep. being the Minimal in terms of number of 1's in the binary representation.
Also known as the lazy Fibonacci representation of n.  Glen Whitney, Oct 21 2017


LINKS

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, apparently unpublished. See Table 2.
Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]


FORMULA



EXAMPLE

As a sum of Fibonacci numbers (A000045) [using 1 at most once], 13 is 13=8+5=8+3+2.
The largest set here is 8+3+2 or, in base Fibonacci, 10110 so a(13)=10110(fib).
The Zeckendorf representation would be the smallest set or {13}=100000(fib).


MAPLE

dualzeckrep:=proc(n)local i, z; z:=zeckrep(n); i:=1; while i<=nops(z)2 do if z[i]=1 and z[i+1]=0 and z[i+2]=0 then z[i]:=0; z[i+1]:=1; z[i+2]:=1; if i>3 then i:=i2 fi else i:=i+1 fi od; if z[1]=0 then z:=subsop(1=NULL, z) fi; z end proc: seq(dualzeckrep(n), n=0..20) ;


MATHEMATICA

fb[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t  Fibonacci[k], AppendTo[fr, 0]]; k]; fr]; a[n_] := Module[{v = fb[n]}, nv = Length[v]; i = 1; While[i <= nv  2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i = 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; 1]]]]]; Array[a, 34, 0] (* Amiram Eldar, Oct 31 2019 after Robert G. Wilson v at A014417 and the Maple code *)
Map[FromDigits, Select[IntegerString[Range[0, 255], 2], StringFreeQ[#, "00"] &]] (* Paolo Xausa, Apr 05 2024 *)


CROSSREFS

A003754 gives the numbers corresponding to the binary digit strings seen here.


KEYWORD

nonn


AUTHOR



EXTENSIONS

Index in formula corrected, missing parts of the maple code recovered, and sequence extended by R. J. Mathar, Oct 23 2010
Definition expanded and Duchêne, Fraenkel et al. reference added by N. J. A. Sloane, Aug 07 2018


STATUS

approved



