cp's OEIS Frontend

A104326 Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00.

%I A104326 #50 Aug 28 2025 08:14:12
%S A104326 0,1,10,11,101,110,111,1010,1011,1101,1110,1111,10101,10110,10111,
%T A104326 11010,11011,11101,11110,11111,101010,101011,101101,101110,101111,
%U A104326 110101,110110,110111,111010,111011,111101,111110,111111,1010101
%N A104326 Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00.
%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 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.
%C A104326 Also known as the lazy Fibonacci representation of n. - _Glen Whitney_, Oct 21 2017
%H A104326 N. J. A. Sloane, <a href="/A104326/b104326.txt">Table of n, a(n) for n = 0..28655</a>
%H A104326 J. L. Brown, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-1/brown.pdf">A new characterization of the Fibonacci numbers</a>, Fibonacci Quarterly 3, no. 1 (1965) 1-8.
%H A104326 Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/WythoffWisdomJune62016.pdf">Wythoff Wisdom</a>, 43 pages, no date, apparently unpublished. See Table 2.
%H A104326 Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, <a href="/A001950/a001950.pdf">Wythoff Wisdom</a>, unpublished, no date [Cached copy, with permission]
%H A104326 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrep.html">Using Fibonacci Numbers to Represent Whole Numbers</a>.
%F A104326 a(n) = A007088(A003754(n+1)).
%e A104326 As a sum of Fibonacci numbers (A000045) [using 1 at most once], 13 is 13=8+5=8+3+2.
%e A104326 The largest set here is 8+3+2 or, in base Fibonacci, 10110 so a(13)=10110(fib).
%e A104326 The Zeckendorf representation 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(n),n=0..20) ;
%p A104326 # alternative
%p A104326 A104326 := proc(n)
%p A104326     local L,itr,rec,i ;
%p A104326     # first compute the usual Zeckendorf rep as in A014417
%p A104326     L := convert(A014417(n),base,10) ;
%p A104326     for itr from 1 do
%p A104326         rec := false ;
%p A104326         # try to recombine 001 -> 110
%p A104326         for i from 3 to nops(L) do
%p A104326             if op(i,L) = 1 and op(i-1,L) =0 and op(i-2,L) =0 then
%p A104326                 rec := true ;
%p A104326                 L := subsop(i=0,L) ;
%p A104326                 L := subsop(i-1=1,L) ;
%p A104326                 L := subsop(i-2=1,L) ;
%p A104326             end if;
%p A104326         end do:
%p A104326         if op(-1,L) = 0 then
%p A104326             L := subsop(-1=NULL,L) ;
%p A104326         end if;
%p A104326         if rec = false then
%p A104326             break ;
%p A104326         end if;
%p A104326     end do:
%p A104326     add( op(i,L)*10^(i-1),i=1..nops(L)) ;
%p A104326 end proc:
%p A104326 seq(A104326(n),n=0..20) ; # _R. J. Mathar_, Aug 28 2025
%t A104326 fb[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr]; a[n_] := Module[{v = fb[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, FromDigits[v[[i[[1, 1]] ;; -1]]]]]; Array[a, 34, 0] (* _Amiram Eldar_, Oct 31 2019 after _Robert G. Wilson v_ at A014417 and the Maple code *)
%t A104326 Map[FromDigits, Select[IntegerString[Range[0, 255], 2], StringFreeQ[#, "00"] &]] (* _Paolo Xausa_, Apr 05 2024 *)
%Y A104326 Cf. A007088 (binary vectors), A014417, A095791, A104324.
%Y A104326 A003754 gives the numbers corresponding to the binary digit strings seen here.
%K A104326 nonn,changed
%O A104326 0,3
%A A104326 _Ron Knott_, Mar 01 2005
%E A104326 Index in formula corrected, missing parts of the maple code recovered, and sequence extended by _R. J. Mathar_, Oct 23 2010
%E A104326 Definition expanded and Duchêne, Fraenkel et al. reference added by _N. J. A. Sloane_, Aug 07 2018