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
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
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