Search: id:A104326
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%I A104326
%S A104326 0,1,10,11,101,110,111,1010,1011,1101,1110,1111,10101,10110,10111,11010,
%T A104326 11011,11101,11110,11111,101010
%N A104326 Dual Zeckendorf representation of n or the maximal (binary) Fibonacci 
               representation.
%C A104326 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.
%D A104326 J L Brown 'A New Characterization of the Fibonacci Numbers' Fibonacci 
               Quarterly, 3 (1965), pp. 1-8
%H A104326 R Knott <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibrep.html">Using Fibonacci Numbers to Represent Whole Numbers</
               a>
%e A104326 As a sum of Fibonacci numbers (A000045) [using 1 at most once],
%e A104326 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)
%p A104326 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
%Y A104326 Cf. A014417, A104324.
%Y A104326 a(n)=A007088(A003754(n)).
%Y A104326 Sequence in context: A125099 A055611 A077813 this_sequence A037090 A118240 
               A157845
%Y A104326 Adjacent sequences: A104323 A104324 A104325 this_sequence A104327 A104328 
               A104329
%K A104326 nonn
%O A104326 0,3
%A A104326 Ron Knott (enquiry(AT)ronknott.com), Mar 01 2005

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