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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 (list; graph; refs; listen; history; text; internal format)
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(n-1).

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

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Last modified April 19 05:18 EDT 2019. Contains 322237 sequences. (Running on oeis4.)