A288175 Positions of 1 in A288173; complement of A288174.

3, 7, 10, 15, 18, 22, 25, 29, 33, 36, 40, 43, 48, 51, 55, 58, 63, 66, 71, 74, 78, 81, 86, 89, 93, 96, 100, 104, 107, 111, 114, 119, 122, 126, 129, 133, 137, 140, 144, 147, 151, 155, 158, 162, 165, 170, 173, 177, 180, 184, 188, 191, 195, 198, 203, 206, 210

Offset

1,1

Comments

Conjecture: lim_{n->infinity} a(n)/n = 3.70..., 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.

We see from this that the first differences of the positions of 1 can be obtained as the image of the sequence t = ABACABABB... under the letter-to-letter morphism lambda given by

lambda(A) = 3, lambda(B) = 4, lambda(C) = 5.

Then

a(n) = 3*N_A(n) + 4*N_B(n) + 5*N_C(n),

where N_X(n) is the number of times the letter X from {A,B,C} occurs in the word t(1)t(2)...t(n).

It follows that a(n)/n is asymptotically equal to the weighted asymptotic frequencies m_A, m_B, m_C of the letters in t:

a(n)/n -> 3*m_A + 4*m_B + 5*m_C.

The existence and values of these frequencies follow from the Perron-Frobenius theorem for nonnegative matrices applied to the incidence matrix of the morphism alpha. This incidence matrix is equal to

|1 1 1 |

|1 0 2 |

|0 1 0 |.

The eigenvalues are cubic irrationals equal to

L1 = 2.17008648..., L2 = 0.3111078169..., L3 = -1.481194304... .

According to the PF-theorem the vector of frequencies (m_A, m_B, m_C) is equal to the normalized eigenvector of the eigenvalue L1

(m_A, m_B, m_C) = (0.46081112715, 0.36910238601, 0.17008648683).

It thus follows that a(n)/n -> 3.7092753596... . (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, A288174.

Sequence in context: A347638 A224880 A043722 * A214066 A120738 A190306

Adjacent sequences: A288172 A288173 A288174 * A288176 A288177 A288178

Keyword

nonn,easy

Author

Clark Kimberling, Jun 07 2017

Status

approved