

A104326


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


21



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

N. J. A. Sloane, Table of n, a(n) for n = 0..28655
J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965) 18.
Eric Duchene, 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Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, apparently unpublished.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Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]
Ron Knott, Using Fibonacci Numbers to Represent Whole Numbers.


FORMULA

a(n) = A007088(A003754(n+1)).


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 *)


CROSSREFS

Cf. A007088 (binary vectors), A014417, A095791, A104324.
Sequence in context: A055611 A077813 A203075 * A205598 A037090 A171676
Adjacent sequences: A104323 A104324 A104325 * A104327 A104328 A104329


KEYWORD

nonn


AUTHOR

Ron Knott, Mar 01 2005


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 Duchene, Fraenkel et al. reference added by N. J. A. Sloane, Aug 07 2018


STATUS

approved



