

A104326


Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation.


7



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.


REFERENCES

J L Brown 'A New Characterization of the Fibonacci Numbers' Fibonacci Quarterly, 3 (1965), pp. 18


LINKS

Table of n, a(n) for n=0..33.
R 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 Zeck. rep. 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) ;


CROSSREFS

Cf. A014417, 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, sequence extended  R. J. Mathar, Oct 23 2010


STATUS

approved



