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%I #8 Aug 29 2020 08:59:03
%S 3,4,8,9,12,13,17,18,22,23,26,27,31,32,35,36,40,41,45,46,49,50,54,55,
%T 59,60,63,64,68,69,72,73,77,78,82,83,86,87,91,92,95,96,100,101,105,
%U 106,109,110,114,115,119,120,123,124,128,129,132,133,137,138,142
%N Positions of 0 in A286749; complement of A286751.
%C a(n) - a(n-1) is in {1,2,3,4} for n>=2, and a(n)/n -> (7 + sqrt(5))/4.
%C From _Michel Dekking_, Aug 29 2020: (Start)
%C There is an explicit expression for the difference sequence Delta a given by Delta a(n) = a(n+1)-a(n).
%C Claim: the sequence Delta a is the decoration a -> 14, b -> 13 of the Fibonacci word abaababaab......
%C Proof: recall from A286749 that A286749 is the letter-to-letter image of the fixed point x of the morphism mu given by
%C mu: 1->12341, 2->1, 3->2, 4->34,
%C where the letter-to-letter map lambda is defined by
%C lambda: 1->1, 2->1, 3->0, 4->0.
%C We see from this that 0's in A286749 correspond uniquely to pairs 34 in x. So we compute the return words of 34. These are 3412 and 34112. Since
%C mu(3412) = 234123411, mu(34112) = 23412341123411,
%C the return words, coded as A = 3412, B = 34112 induce a descendant morphism
%C A->AB, B->ABB.
%C This well-known morphism (see A096270) has the property that its unique fixed point is the Fibonacci word (on the alphabet {B,A}), preceded by the letter A.
%C The return word 3412 has lambda-image 0011, and the return word 34112 has lambda-image 00111. This means that they give distances 1 and 3, respectively 1 and 4 between (successive) occurrences of 0's in A286749.
%C This leads to the decoration A->13, B->14.
%C That a(n)/n -> (7 + sqrt(5))/4 follows from the corresponding result for the sequence A286751.
%C (End)
%H Clark Kimberling, <a href="/A286750/b286750.txt">Table of n, a(n) for n = 1..10000</a>
%e As a word, A286749 = 11001110011001110011010..., in which 0 is in positions 3,4,8,9,12,...
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 12]; (* A003849 *)
%t w = StringJoin[Map[ToString, s]];
%t w1 = StringReplace[w, {"0100" -> ""}]; st = ToCharacterCode[w1] - 48; (* A286749 *)
%t Flatten[Position[st, 0]]; (* A286750 *)
%t Flatten[Position[st, 1]]; (* A286751 *)
%Y Cf. A003849, A286749, A286751.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, May 14 2017
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