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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A164349 The limit of the string "0, 1" under the operation 'repeat string twice and remove last symbol'.

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%I A164349 #22 Aug 02 2015 02:07:45
%S A164349 0,1,0,0,1,0,1,0,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,0,1,0,0,1,0,1,0,
%T A164349 1,0,0,1,0,1,0,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,0,1,0,0,1,0,0,1,0,
%U A164349 0,1,0,1,0,0,0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,0,1,0,0,1,0,1,0,1,0,0,1,0,1
%N A164349 The limit of the string "0, 1" under the operation 'repeat string twice and remove last symbol'.
%C A164349 We start with the string 01, at each step we replace the string by two concatenated copies and remove the last symbol.
%C A164349 01 -> 010 -> 01001 -> 010010100 etc.
%C A164349 Each string consists of 2^n + 1 symbols and clearly after this step the first 2^n + 1 symbols do not change.
%C A164349 Equivalently this sequence is given as follows: a(0) = 0, a(1) = 1, and for n>1, a(n) = first one of g(n), g(g(n)), g(g(g(n))).. etc. to be either 0 or 1, where g(n) = A053645(n-1).
%C A164349 The proportion of 0's in this sequence converges to a number close to 0.645059. The exact nature of this constant is not known. It is easy to show there are no 2 consecutive 1's.
%C A164349 Start of the first occurrence of k consecutive zeros: 0, 2, 7, 2046, > 8388600, ..., . - _Robert G. Wilson v_, Aug 17 2009
%C A164349 Start of the first occurrence of 5 consecutive zeros is > 2^34 - 5. Sum of the first 10^n terms b(n) begins: 0, 3, 36, 355, 3549, 35494, 354942, 3549412, 35494122, 354941215, 3549412151. - _Alex Ratushnyak_, Aug 15 2012
%C A164349 a(A246439(n)) = 1; a(A246438(n)) = 0. - _Reinhard Zumkeller_, Aug 28 2014
%C A164349 The partial sums appear to give A101402. - _Arie Groeneveld_, Aug 27 2014
%H A164349 Robert G. Wilson v, <a href="/A164349/b164349.txt">Table of n, a(n) for n = 0..16384 </a>
%t A164349 Nest[ Most@ Flatten@ {#, #} &, {0, 1}, 7] (* _Robert G. Wilson v_, Aug 17 2009 *)
%o A164349 (Perl) my $ab = "10"; for (my $j = 1; $j < 30; $j++) { $ab .= $ab; substr $ab, -1, 1, ""; print "$ab\n"; }
%o A164349 (Haskell)
%o A164349 a164349 n = if n == 0 then 0 else until (<= 1) (a053645 . subtract 1) n
%o A164349 -- _Reinhard Zumkeller_, Aug 28 2014
%Y A164349 Cf. A053645, A246438, A246439, A101402.
%K A164349 easy,nonn
%O A164349 0,1
%A A164349 _Jack W Grahl_, Aug 13 2009
%E A164349 Spelling and notation corrected by _Charles R Greathouse IV_, Mar 23 2010