



3, 4, 8, 9, 12, 13, 17, 18, 22, 23, 26, 27, 31, 32, 35, 36, 40, 41, 45, 46, 49, 50, 54, 55, 59, 60, 63, 64, 68, 69, 72, 73, 77, 78, 82, 83, 86, 87, 91, 92, 95, 96, 100, 101, 105, 106, 109, 110, 114, 115, 119, 120, 123, 124, 128, 129, 132, 133, 137, 138, 142
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OFFSET

1,1


COMMENTS

a(n)  a(n1) is in {1,2,3,4} for n>=2, and a(n)/n > (7 + sqrt(5))/4.
From Michel Dekking, Aug 29 2020: (Start)
There is an explicit expression for the difference sequence Delta a given by Delta a(n) = a(n+1)a(n).
Claim: the sequence Delta a is the decoration a > 14, b > 13 of the Fibonacci word abaababaab......
Proof: recall from A286749 that A286749 is the lettertoletter image of the fixed point x of the morphism mu given by
mu: 1>12341, 2>1, 3>2, 4>34,
where the lettertoletter map lambda is defined by
lambda: 1>1, 2>1, 3>0, 4>0.
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
mu(3412) = 234123411, mu(34112) = 23412341123411,
the return words, coded as A = 3412, B = 34112 induce a descendant morphism
A>AB, B>ABB.
This wellknown 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.
The return word 3412 has lambdaimage 0011, and the return word 34112 has lambdaimage 00111. This means that they give distances 1 and 3, respectively 1 and 4 between (successive) occurrences of 0's in A286749.
This leads to the decoration A>13, B>14.
That a(n)/n > (7 + sqrt(5))/4 follows from the corresponding result for the sequence A286751.
(End)


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000


EXAMPLE

As a word, A286749 = 11001110011001110011010..., in which 0 is in positions 3,4,8,9,12,...


MATHEMATICA

s = Nest[Flatten[# /. {0 > {0, 1}, 1 > {0}}] &, {0}, 12]; (* A003849 *)
w = StringJoin[Map[ToString, s]];
w1 = StringReplace[w, {"0100" > ""}]; st = ToCharacterCode[w1]  48; (* A286749 *)
Flatten[Position[st, 0]]; (* A286750 *)
Flatten[Position[st, 1]]; (* A286751 *)


CROSSREFS

Cf. A003849, A286749, A286751.
Sequence in context: A285033 A073258 A170954 * A135135 A058593 A310017
Adjacent sequences: A286747 A286748 A286749 * A286751 A286752 A286753


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 14 2017


STATUS

approved



