A287105 Positions of 0 in A287104.

3, 5, 9, 12, 16, 19, 21, 24, 28, 31, 33, 37, 40, 42, 45, 49, 52, 54, 58, 61, 65, 68, 70, 74, 77, 79, 82, 86, 89, 91, 95, 98, 102, 105, 107, 110, 114, 117, 119, 123, 126, 130, 133, 135, 139, 142, 144, 147, 151, 154, 156, 160, 163, 167, 170, 172, 175, 179, 182

Offset: 1

Author

Clark Kimberling, May 21 2017

Keywords

• nonn
• easy

Comments

From Michel Dekking, Sep 17 2019: (Start)

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

Let u=01, v=012, w=0121 be the return words of the word 0.

[See Justin & Vuillon (2000) for definition of return word. - N. J. A. Sloane, Sep 23 2019]

Then under sigma u, v and w are mapped to

sigma(01) = 1012, sigma(012) = 10120, sigma(0121) = 1012012.

Moving the prefix 1 of these three images to the end, the sequence 0 a (i.e., (a(n)) prefixed by the symbol 0), is a fixed point when iterating.

This iteration process induces a morphism 2->4, 3->32, 4->34 on the return words, coded by their lengths.

Coding the symbols according to 2<->2, 4<->0, 3<->1, this leads to the morphism 2->0, 1->12, 0->10 on the alphabet {0,1,2}.

This is simply sigma, which has A287104 as its unique fixed point. So the sequence d of first differences of (a(n)) equals A287104 with the coding above. Noting that the code can be written as x->4-x, 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, A287106, A287107.

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) = 4n-1 + Sum_{k=2..n} A287104(k). - Michel Dekking, Sep 17 2019