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A104326 Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00. 9
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 (list; graph; refs; listen; history; text; internal format)



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


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) 1-8.

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Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, apparently unpublished.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Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]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


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


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).


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:=i-2 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) ;


Cf. A007088 (binary vectors), A014417, A095791, A104324.

Sequence in context: A055611 A077813 A203075 * A205598 A037090 A171676

Adjacent sequences:  A104323 A104324 A104325 * A104327 A104328 A104329




Ron Knott, Mar 01 2005


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



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Last modified January 15 18:48 EST 2019. Contains 319170 sequences. (Running on oeis4.)