

A164349


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


10



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, 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, 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
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OFFSET

0,1


COMMENTS

We start with the string 01, at each step we replace the string by two concatenated copies and remove the last symbol.
01 > 010 > 01001 > 010010100 etc.
Each string consists of 2^n + 1 symbols and clearly after this step the first 2^n + 1 symbols do not change.
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(n1).
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.
Start of the first occurrence of k consecutive zeros: 0, 2, 7, 2046, > 8388600, ..., .  Robert G. Wilson v, Aug 17 2009
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
a(A246439(n)) = 1; a(A246438(n)) = 0.  Reinhard Zumkeller, Aug 28 2014
The partial sums appear to give A101402.  Arie Groeneveld, Aug 27 2014


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..16384


MATHEMATICA

Nest[ Most@ Flatten@ {#, #} &, {0, 1}, 7] (* Robert G. Wilson v, Aug 17 2009 *)


PROG

(Perl) my $ab = "10"; for (my $j = 1; $j < 30; $j++) { $ab .= $ab; substr $ab, 1, 1, ""; print "$ab\n"; }
(Haskell)
a164349 n = if n == 0 then 0 else until (<= 1) (a053645 . subtract 1) n
 Reinhard Zumkeller, Aug 28 2014


CROSSREFS

Cf. A053645, A246438, A246439, A101402.
Sequence in context: A091445 A091446 A270742 * A094186 A267371 A285205
Adjacent sequences: A164346 A164347 A164348 * A164350 A164351 A164352


KEYWORD

easy,nonn


AUTHOR

Jack W Grahl, Aug 13 2009


EXTENSIONS

Spelling and notation corrected by Charles R Greathouse IV, Mar 23 2010


STATUS

approved



