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%I #12 Feb 23 2020 14:17:06
%S 1,2,4,5,6,8,9,11,12,13,14,16,17,19,20,21,23,24,26,27,28,30,31,32,34,
%T 35,37,38,39,41,42,44,45,46,47,49,50,52,53,54,56,57,59,60,61,62,64,65,
%U 67,68,69,70,72,73,75,76,77,79,80,82,83,84,85,87,88,90
%N Positions of 0 in A288173; complement of A288175.
%C Conjecture: lim_{n->infinity} a(n)/n = 1.36..., and if m denotes this number, then -1 < m - a(n)/n < 1 for n >= 1.
%C From _Michel Dekking_, Feb 23 2020: (Start)
%C Proof of the first part of this conjecture.
%C Let a(0):=0. We write this sequence as the sum of its first differences:
%C a(n) = Sum_{k=0..n-1} a(k+1)-a(k).
%C We know (see A288173) that A288173 can be generated as a decoration delta(t) of the fixed point t of the morphism alpha given by
%C alpha(A) = AB, alpha(B) = AC, alpha(C) = ABB.
%C Here delta is the morphism
%C delta(A) = 001, delta(B) = 0001, delta(C) = 00001.
%C Let e = A288175 be the sequence of positions of 1 in A288173. Note that if we are at the n-th 1, then we have seen e(n)-n zeros. So the position of the (e(n)-n)-th zero is e(n)-1.
%C Let m(n):=e(n)-n. Then
%C a(m(n))/m(n) = (e(n)-1)/m(n) = (e(n)-1)/n * n/(e(n)-n).
%C According to the comments at e = A288175, the first factor in this product converges to 3.7092753596..., and the second to 1/(3.7092753596... - 1).
%C It follows that as n->infinity,
%C a(m(n))/m(n) -> 1.36900369004... .
%C It is easy to see from this that the whole sequence converges, and so
%C a(n)/n -> 1.36900369004... . (End)
%H Clark Kimberling, <a href="/A288174/b288174.txt">Table of n, a(n) for n = 1..10000</a>
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "1" -> "001"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[11]] - 48 (* A288173 *)
%t Flatten[Position[st, 0]] (* A288174 *)
%t Flatten[Position[st, 1]] (* A288175 *)
%Y Cf. A288173, A288175.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jun 07 2017