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 A104326 Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00. 21
 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)
 OFFSET 0,3 COMMENTS 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. Also known as the lazy Fibonacci representation of n. - Glen Whitney, Oct 21 2017 LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..28655 J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, no. 1 (1965) 1-8. Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, apparently unpublished. See Table 2. Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission] FORMULA a(n) = A007088(A003754(n+1)). EXAMPLE 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). MAPLE 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=0 then z:=subsop(1=NULL, z) fi; z end proc: seq(dualzeckrep(n), n=0..20) ; MATHEMATICA fb[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt]], 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 *) CROSSREFS Cf. A007088 (binary vectors), A014417, A095791, A104324. Sequence in context: A055611 A077813 A203075 * A205598 A037090 A171676 Adjacent sequences:  A104323 A104324 A104325 * A104327 A104328 A104329 KEYWORD nonn AUTHOR Ron Knott, Mar 01 2005 EXTENSIONS Index in formula corrected, missing parts of the maple code recovered, and sequence extended by R. J. Mathar, Oct 23 2010 Definition expanded and Duchene, Fraenkel et al. reference added by N. J. A. Sloane, Aug 07 2018 STATUS approved

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Last modified February 27 12:47 EST 2020. Contains 332306 sequences. (Running on oeis4.)