A286750 Positions of 0 in A286749; complement of A286751.

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

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 * A359376 A135135 A058593

Adjacent sequences: A286747 A286748 A286749 * A286751 A286752 A286753

Keyword

nonn,easy

Author

Clark Kimberling, May 14 2017

Status

approved