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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 (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.


Table of n, a(n) for n=0..33.

J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965) 1-8.

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



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Last modified February 12 21:30 EST 2016. Contains 268229 sequences.