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A329101 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the base 4 expansion of n equals the number of 2's in the base 4 expansion of a(n).

%I A329101 #24 Nov 08 2019 09:46:49
%S A329101 0,2,1,3,6,10,8,9,4,11,5,7,12,14,13,15,18,26,22,24,34,42,38,40,25,41,
%T A329101 27,30,32,43,33,35,16,36,17,19,37,46,39,44,20,45,21,23,28,47,29,31,48,
%U A329101 50,49,51,54,58,56,57,52,59,53,55,60,62,61,63,66,74,70,72
%N A329101 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the base 4 expansion of n equals the number of 2's in the base 4 expansion of a(n).
%C A329101 This sequence is a permutation of the nonnegative integers with inverse A329180.
%C A329101 Apparently, fixed points correspond to A001196.
%C A329101 The sequence has fractal features; for any k >= 0, the set of points { (n, a(n)), n = 0..4^k-1 } is symmetrical relative to the line of equation y + x = 4^k - 1 (see scatterplots in Links section).
%H A329101 Rémy Sigrist, <a href="/A329101/b329101.txt">Table of n, a(n) for n = 0..4095</a>
%H A329101 Rémy Sigrist, <a href="/A329101/a329101_1.gp.txt">PARI program for A329101</a>
%H A329101 Rémy Sigrist, <a href="/A329101/a329101.png">Scatterplot of the sequence for n = 0..4^3-1</a>
%H A329101 Rémy Sigrist, <a href="/A329101/a329101_1.png">Scatterplot of the sequence for n = 0..4^10-1</a>
%H A329101 Rémy Sigrist, <a href="/A329101/a329101_2.png">Colored scatterplot of the sequence for n = 0..4^10-1</a> (where the color is function of A160381(n))
%H A329101 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A329101 A160381(n) = A160382(a(n)).
%e A329101 The first terms, alongside the base 4 representations of n and of a(n), are:
%e A329101   n   a(n)  qua(n)  qua(a(n))
%e A329101   --  ----  ------  ---------
%e A329101    0     0       0          0
%e A329101    1     2       1          2
%e A329101    2     1       2          1
%e A329101    3     3       3          3
%e A329101    4     6      10         12
%e A329101    5    10      11         22
%e A329101    6     8      12         20
%e A329101    7     9      13         21
%e A329101    8     4      20         10
%e A329101    9    11      21         23
%e A329101   10     5      22         11
%e A329101   11     7      23         13
%e A329101   12    12      30         30
%e A329101   13    14      31         32
%e A329101   14    13      32         31
%e A329101   15    15      33         33
%o A329101 (PARI) See Links section.
%Y A329101 Cf. A001196, A160381, A160382, A329180.
%K A329101 nonn,base
%O A329101 0,2
%A A329101 _Rémy Sigrist_, Nov 07 2019