A288174 - OEIS

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

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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"}]