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Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00.
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%I #46 Apr 05 2024 13:08:12

%S 0,1,10,11,101,110,111,1010,1011,1101,1110,1111,10101,10110,10111,

%T 11010,11011,11101,11110,11111,101010,101011,101101,101110,101111,

%U 110101,110110,110111,111010,111011,111101,111110,111111,1010101

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

%C 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 Also known as the lazy Fibonacci representation of n. - _Glen Whitney_, Oct 21 2017

%H N. J. A. Sloane, <a href="/A104326/b104326.txt">Table of n, a(n) for n = 0..28655</a>

%H J. L. Brown, Jr., <a href="http://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 Eric DuchĂȘne, 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, 43 pages, no date, apparently unpublished.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 Eric DuchĂȘne, 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, unpublished, no date [Cached copy, with permission]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 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 a(n) = A007088(A003754(n+1)).

%e As a sum of Fibonacci numbers (A000045) [using 1 at most once], 13 is 13=8+5=8+3+2.

%e The largest set here is 8+3+2 or, in base Fibonacci, 10110 so a(13)=10110(fib).

%e The Zeckendorf representation would be the smallest set or {13}=100000(fib).

%p 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) ;

%t 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 Map[FromDigits, Select[IntegerString[Range[0, 255], 2], StringFreeQ[#, "00"] &]] (* _Paolo Xausa_, Apr 05 2024 *)

%Y Cf. A007088 (binary vectors), A014417, A095791, A104324.

%Y A003754 gives the numbers corresponding to the binary digit strings seen here.

%K nonn

%O 0,3

%A _Ron Knott_, Mar 01 2005

%E Index in formula corrected, missing parts of the maple code recovered, and sequence extended by _R. J. Mathar_, Oct 23 2010

%E Definition expanded and DuchĂȘne, Fraenkel et al. reference added by _N. J. A. Sloane_, Aug 07 2018