

A164349


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


9



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: A051065 A091445 A091446 * A094186 A003849 A115199
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



