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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 07 2017
STATUS
approved