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A372659 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n do not appear in the dual Zeckendorf representation of a(n).

%I A372659 #9 May 12 2024 11:25:05
%S A372659 0,2,1,3,20,4,15,12,5,7,13,8,29,6,10,21,16,36,9,19,63,11,18,17,28,33,
%T A372659 14,26,59,22,54,56,57,101,23,34,25,27,96,46,53,88,24,44,51,42,211,38,
%U A372659 49,93,92,180,47,91,207,30,37,64,50,62,43,60,80,31,41,85,76
%N A372659 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n do not appear in the dual Zeckendorf representation of a(n).
%C A372659 The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
%C A372659 This sequence is a permutation of the nonnegative integers with inverse A372660.
%H A372659 Rémy Sigrist, <a href="/A372659/b372659.txt">Table of n, a(n) for n = 0..10000</a>
%H A372659 Rémy Sigrist, <a href="/A372659/a372659.gp.txt">PARI program</a>
%H A372659 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>
%H A372659 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A372659 The first terms, alongside the Zeckendorf representation of n and the dual Zeckendorf representation of a(n), in binary, are:
%e A372659   n   a(n)  z(n)    d(a(n))
%e A372659   --  ----  ------  --------
%e A372659    0     0       0         0
%e A372659    1     2       1        10
%e A372659    2     1      10         1
%e A372659    3     3     100        10
%e A372659    4    20     101    101010
%e A372659    5     4    1000       101
%e A372659    6    15    1001    110110
%e A372659    7    12    1010     10101
%e A372659    8     5   10000       111
%e A372659    9     7   10001      1110
%e A372659   10    13   10010    101101
%e A372659   11     8   10100      1011
%e A372659   12    29   10101  10101010
%o A372659 (PARI) \\ See Links section.
%Y A372659 See A372657 for a similar sequence.
%Y A372659 Cf. A356771, A372660 (inverse).
%K A372659 nonn,base
%O A372659 0,2
%A A372659 _Rémy Sigrist_, May 09 2024