A287106 Positions of 1 in A287104.

1, 4, 6, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 36, 38, 41, 43, 46, 48, 50, 53, 55, 57, 59, 62, 64, 66, 69, 71, 73, 75, 78, 80, 83, 85, 87, 90, 92, 94, 96, 99, 101, 103, 106, 108, 111, 113, 115, 118, 120, 122, 124, 127, 129, 131, 134, 136, 138, 140

Offset: 1

Author

Clark Kimberling, May 21 2017

Keywords

• nonn
• easy

Comments

From Michel Dekking, Sep 16 2019: (Start)

Let sigma be the defining morphism in A287104: 0->10, 1->12, 2->0.

Let u := 10, v := 12, w: = 120 be the return words of the word 1. [See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]

Then

sigma(u) = vu, sigma(v) = w, sigma(w) = wu.

If we code w<->0, u<->1, v<->2, then this morphism turns into the morphism

0 -> 01, 1 -> 21, 2 -> 0.

This is exactly the morphism which has A287072 as unique fixed point.

Since u and v have length 2 and w has length 3, this implies that the sequence d of first differences of (a(n)) equals A287072 with the projection 0 -> 3, 1 -> 2, 2 -> 2. This gives the formula below.

(End)

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Jacques Justin and Laurent Vuillon, Return words in Sturmian and episturmian words, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.

Crossrefs

Cf. A287104, A287105, A287107. Closely related to A287072.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 2}, 2 -> 0}] &, {0}, 10] (* A287104 *)
Flatten[Position[s, 0]] (* A287105 *)
Flatten[Position[s, 1]] (* A287106 *)
Flatten[Position[s, 2]] (* A287107 *)

Formula

a(n) = 1 + Sum_{k=1..n-1} d(k), where d(k) = 3 if A287072(k)=0, and d(k) = 2 otherwise, for k = 1,...,n. - Michel Dekking, Sep 16 2019