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A372657 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n are not missing from the dual Zeckendorf representation of a(n).

%I A372657 #13 May 12 2024 11:24:47
%S A372657 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,19,16,21,18,20,22,23,24,25,
%T A372657 28,29,31,26,27,32,30,33,34,35,36,38,40,37,42,39,46,48,47,51,53,41,43,
%U A372657 44,45,56,49,50,52,54,55,57,58,59,62,63,65,60,61,66,64,67
%N A372657 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n are not missing from the dual Zeckendorf representation of a(n).
%C A372657 We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.
%C A372657 The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
%C A372657 This sequence is a permutation of the nonnegative integers with inverse A372658: for any v >= 0, the majority of Fibonacci numbers are not missing from the dual Zeckendorf representation of v, and provide opportunities for v to be chosen, and so v will eventually appear in the sequence.
%H A372657 Rémy Sigrist, <a href="/A372657/b372657.txt">Table of n, a(n) for n = 0..10000</a>
%H A372657 Rémy Sigrist, <a href="/A372657/a372657.gp.txt">PARI program</a>
%H A372657 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>
%H A372657 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A372657 The first terms, alongside the Zeckendorf representation of n and the dual Zeckendorf representation of a(n), in binary, are:
%e A372657   n   a(n)  z(n)    d(a(n))
%e A372657   --  ----  ------  -------
%e A372657    0     0       0        0
%e A372657    1     1       1        1
%e A372657    2     2      10       10
%e A372657    3     3     100       11
%e A372657    4     4     101      101
%e A372657    5     5    1000      110
%e A372657    6     6    1001      111
%e A372657    7     7    1010     1010
%e A372657    8     8   10000     1011
%e A372657    9     9   10001     1101
%e A372657   10    10   10010     1110
%e A372657   11    11   10100     1111
%e A372657   12    12   10101    10101
%o A372657 (PARI) \\ See Links section.
%Y A372657 See A372659 for a similar sequence.
%Y A372657 Cf. A356771, A361989, A372658 (inverse).
%K A372657 nonn,base
%O A372657 0,3
%A A372657 _Rémy Sigrist_, May 09 2024