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A367262 Lexicographically earliest sequence of distinct nonnegative integers such that the values a(0) XOR ... XOR a(k) (for some k >= 0) are all distinct (where XOR denotes the bitwise XOR operator).

%I A367262 #14 Nov 13 2023 11:31:56
%S A367262 0,1,2,4,3,6,7,8,5,14,9,16,10,11,12,15,13,25,17,18,19,21,22,20,24,29,
%T A367262 23,32,26,27,28,30,31,49,33,35,34,37,38,41,40,36,44,64,39,42,43,45,46,
%U A367262 47,48,50,51,52,54,53,56,55,59,57,60,58,66,65,69,67,61,96
%N A367262 Lexicographically earliest sequence of distinct nonnegative integers such that the values a(0) XOR ... XOR a(k) (for some k >= 0) are all distinct (where XOR denotes the bitwise XOR operator).
%C A367262 This sequence is a variant of A333400; here we combine initial terms with the XOR operator, there with the addition.
%C A367262 This sequence is well defined; after some initial terms we can extend the sequence with a power of 2 greater that any prior term or even a smaller value.
%C A367262 This sequence is a permutation of the nonnegative integers (with inverse A367263):
%C A367262 - for any k >= 0, the least value >= 2^k is precisely 2^k,
%C A367262 - all powers of 2 appear in the sequence,
%C A367262 - after a power of 2, if the least value not yet in the sequence is less than this power of 2, then this value will be the next term.
%H A367262 Rémy Sigrist, <a href="/A367262/b367262.txt">Table of n, a(n) for n = 0..10000</a>
%H A367262 Rémy Sigrist, <a href="/A367262/a367262.txt">C++ program</a>
%H A367262 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A367262 The first terms are:
%e A367262   n   a(n)  a(0) XOR ... XOR a(n)
%e A367262   --  ----  ---------------------
%e A367262    0     0                      0
%e A367262    1     1                      1
%e A367262    2     2                      3
%e A367262    3     4                      7
%e A367262    4     3                      4
%e A367262    5     6                      2
%e A367262    6     7                      5
%e A367262    7     8                     13
%e A367262    8     5                      8
%e A367262    9    14                      6
%e A367262   10     9                     15
%e A367262   11    16                     31
%e A367262   12    10                     21
%o A367262 (C++) See Links section.
%Y A367262 Cf. A333400, A346298, A367263 (inverse), A367264.
%K A367262 nonn,base
%O A367262 0,3
%A A367262 _Rémy Sigrist_, Nov 11 2023