0,3

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 Represntation 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.

J L Brown 'A New Characterization of the Fibonacci Numbers' Fibonacci Quarterly, 3 (1965), pp. 1-8

R Knott Using Fibonacci Numbers to Represent Whole Numbers

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)

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

Cf. A014417, A104324.

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

Sequence in context: A125099 A055611 A077813 this_sequence A037090 A118240 A157845

Adjacent sequences: A104323 A104324 A104325 this_sequence A104327 A104328 A104329

nonn

Ron Knott (enquiry(AT)ronknott.com), Mar 01 2005