A288174 Positions of 0 in A288173; complement of A288175.

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, 35, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 84, 85, 87, 88, 90

Offset

1,2

Comments

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.

From Michel Dekking, Feb 23 2020: (Start)

Proof of the first part of this conjecture.

Let a(0):=0. We write this sequence as the sum of its first differences:

a(n) = Sum_{k=0..n-1} a(k+1)-a(k).

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

alpha(A) = AB, alpha(B) = AC, alpha(C) = ABB.

Here delta is the morphism

delta(A) = 001, delta(B) = 0001, delta(C) = 00001.

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.

Let m(n):=e(n)-n. Then

a(m(n))/m(n) = (e(n)-1)/m(n) = (e(n)-1)/n * n/(e(n)-n).

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).

It follows that as n->infinity,

a(m(n))/m(n) -> 1.36900369004... .

It is easy to see from this that the whole sequence converges, and so

a(n)/n -> 1.36900369004... . (End)

Links

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

Mathematica

s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "1" -> "001"}]
Table[w[n], {n, 0, 8}]
st = ToCharacterCode[w[11]] - 48   (* A288173 *)
Flatten[Position[st, 0]]  (* A288174 *)
Flatten[Position[st, 1]]  (* A288175 *)

Crossrefs

Cf. A288173, A288175.

Sequence in context: A328594 A346129 A366160 * A280998 A043687 A087118

Adjacent sequences: A288171 A288172 A288173 * A288175 A288176 A288177

Keyword

nonn,easy

Author

Clark Kimberling, Jun 07 2017

Status

approved