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A286750 Positions of 0 in A286749; complement of A286751. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) - a(n-1) 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 letter-to-letter image of the fixed point x of the morphism mu given by

      mu:  1->12341, 2->1, 3->2, 4->34,

where the letter-to-letter 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 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.

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.

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

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Last modified June 12 10:58 EDT 2021. Contains 344947 sequences. (Running on oeis4.)