

A287105


Positions of 0 in A287104.


4



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
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OFFSET

1,1


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>4x, 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, RAIROTheoretical Informatics and Applications 34.5 (2000): 343356.


FORMULA

a(n) = 4n1 + Sum_{k=2..n} A287104(k).  Michel Dekking, Sep 17 2019


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 *)


CROSSREFS

Cf. A287104, A287106, A287107.
Sequence in context: A091785 A191403 A233779 * A003075 A061562 A006282
Adjacent sequences: A287102 A287103 A287104 * A287106 A287107 A287108


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 21 2017


STATUS

approved



