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A347196 Let c(k) be the infinite binary string 010111001... (A030308), the concatenation of reverse order integer binary words ( 0;1;01;11;001;101;... ). a(n) is the bit index k of the first occurrence of the reverse order binary word of n ( n = 2^0*c(a(n)) + 2^1*c(a(n)+1) + ... ).

%I A347196 #35 Oct 09 2021 15:55:57
%S A347196 0,1,0,3,6,1,2,3,18,5,0,8,6,1,2,13,50,17,32,4,23,9,7,29,18,5,0,37,34,
%T A347196 1,12,13,130,49,88,16,67,31,20,3,56,22,24,8,6,38,28,84,50,17,32,4,70,
%U A347196 9,39,36,90,33,0,40,110,11,12,43,322,129,224,48,175,87,53,15
%N A347196 Let c(k) be the infinite binary string 010111001... (A030308), the concatenation of reverse order integer binary words ( 0;1;01;11;001;101;... ). a(n) is the bit index k of the first occurrence of the reverse order binary word of n ( n = 2^0*c(a(n)) + 2^1*c(a(n)+1) + ... ).
%C A347196 It is not surprising to see dyadic self-similarity in the graph of this sequence. For example the graph of a(0..2^9) looks like a rescaled version of a(0..2^8). Each of these intervals reminds a bit of particle traces in a cloud chamber.
%H A347196 Thomas Scheuerle, <a href="/A347196/b347196.txt">Table of n, a(n) for n = 0..5000</a>
%F A347196 a(n) <= Sum_{k=0..n} A070939(k).
%e A347196 pos:0,1,2,3,4,5,6,7,8,9,...
%e A347196 c:  0|1|0,1|1,1|0,0,1|1,0,1...
%e A347196     0                           a(0) = 0
%e A347196     . 1                         a(1) = 1
%e A347196     0 1                         a(2) = 0
%e A347196     . . . 1 1                   a(3) = 3
%e A347196     . . . . . . 0 0 1           a(4) = 6
%e A347196     . 1 0 1                     a(5) = 1
%e A347196     . . 0 1 1                   a(6) = 2
%o A347196 (MATLAB)
%o A347196 function a = A347196( max_n)
%o A347196     c = 0; a = 0;
%o A347196     for n = 1:max_n
%o A347196         b = bitget(n,1:64);
%o A347196         c = [c b(1:find(b == 1, 1, 'last' ))];
%o A347196     end
%o A347196     for n = 1:max_n
%o A347196         b = bitget(n,1:64);
%o A347196         word = b(1:find(b == 1, 1, 'last' ));
%o A347196         pos = strfind(c, word);
%o A347196         a(n+1) = pos(1)-1;
%o A347196     end
%o A347196 end
%Y A347196 Cf. A030304, A030308, A030311, A070939.
%K A347196 nonn,base,easy,look
%O A347196 0,4
%A A347196 _Thomas Scheuerle_, Aug 22 2021