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A372654 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the dual Zeckendorf representations of n and a(n) have no common missing Fibonacci number.

%I A372654 #13 May 12 2024 11:24:24
%S A372654 0,1,3,2,5,4,6,9,10,7,8,11,15,16,17,12,13,14,19,18,25,26,27,29,28,20,
%T A372654 21,22,24,23,31,30,32,41,42,43,45,44,47,46,48,33,34,35,37,36,39,38,40,
%U A372654 51,52,49,50,53,67,68,69,71,70,73,72,74,77,78,75,76,79,54
%N A372654 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the dual Zeckendorf representations of n and a(n) have no common missing Fibonacci number.
%C A372654 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 A372654 The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
%C A372654 This sequence is a self-inverse permutation of the nonnegative integers.
%H A372654 Rémy Sigrist, <a href="/A372654/b372654.txt">Table of n, a(n) for n = 0..10944</a>
%H A372654 Rémy Sigrist, <a href="/A372654/a372654_1.png">Scatterplot of the sequence for n = 0..28655</a>
%H A372654 Rémy Sigrist, <a href="/A372654/a372654.png">Scatterplot of (x, y) such that the dual Zeckendorf representations of x and y have no common missing term and x, y <= 1595</a>
%H A372654 Rémy Sigrist, <a href="/A372654/a372654.gp.txt">PARI program</a>
%H A372654 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>
%H A372654 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A372654 The first terms, alongside the dual Zeckendorf representation in binary of n and of a(n), are:
%e A372654   n   a(n)  z(n)   z(a(n))
%e A372654   --  ----  -----  -------
%e A372654    0     0      0        0
%e A372654    1     1      1        1
%e A372654    2     3     10       11
%e A372654    3     2     11       10
%e A372654    4     5    101      110
%e A372654    5     4    110      101
%e A372654    6     6    111      111
%e A372654    7     9   1010     1101
%e A372654    8    10   1011     1110
%e A372654    9     7   1101     1010
%e A372654   10     8   1110     1011
%e A372654   11    11   1111     1111
%e A372654   12    15  10101    11010
%e A372654   13    16  10110    11011
%o A372654 (PARI) \\ See Links section.
%Y A372654 See A332022 for a similar sequence.
%Y A372654 Cf. A356771, A361989, A372655.
%K A372654 nonn,base
%O A372654 0,3
%A A372654 _Rémy Sigrist_, May 09 2024