A288175 - OEIS

%I #11 Feb 23 2020 14:17:16

%S 3,7,10,15,18,22,25,29,33,36,40,43,48,51,55,58,63,66,71,74,78,81,86,

%T 89,93,96,100,104,107,111,114,119,122,126,129,133,137,140,144,147,151,

%U 155,158,162,165,170,173,177,180,184,188,191,195,198,203,206,210

%N Positions of 1 in A288173; complement of A288174.

%C 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.

%C From _Michel Dekking_, Feb 23 2020: (Start)

%C Proof of the first part of this conjecture.

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

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

%C 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

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

%C Here delta is the morphism

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

%C 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

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

%C Then

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

%C 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).

%C 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:

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

%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

%C       |1 1 1 |

%C       |1 0 2 |

%C       |0 1 0 |.

%C The eigenvalues are cubic irrationals equal to

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

%C 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

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

%C It thus follows that a(n)/n -> 3.7092753596... . (End)

%H Clark Kimberling, <a href="/A288175/b288175.txt">Table of n, a(n) for n = 1..10000</a>

%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];

%t w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "1" -> "001"}]

%t Table[w[n], {n, 0, 8}]

%t st = ToCharacterCode[w[11]] - 48   (* A288173 *)

%t Flatten[Position[st, 0]]  (* A288174 *)

%t Flatten[Position[st, 1]]  (* A288175 *)

%Y Cf. A288173, A288174.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 07 2017