A relative of the Thue-Morse sequence

Allouche, Jean‐Paul; Arnold, André; Berstel, Jean; Brlek, Srečko; Jockusch, William; Plouffe, Simon; Sagan, Bruce E.

doi:10.1016/0012-365x(93)00147-w

Cited by 32 publications

(47 citation statements)

References 3 publications

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“…Observe that T 2 = {2 2i (2j + 1): i, j ∈ N 0 }. Theorem 6 with k = 2 was proved by Allouche et al [3], who also proved that T 2 has the maximum density out of all 2-multiplicative sets. Interesting relationships with the Thue-Morse sequence were also discovered.…”

Section: An Improved Constructionmentioning

confidence: 89%

Colourings of the cartesian product of graphs and multiplicative Sidon sets

Pór

Wood

2009

Combinatorica

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Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is F-free. The F-free chromatic number χ(G, F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies F-free colourings of the cartesian product of graphs.Let H be the cartesian product of the graphs G1, G2, . . . , G d . Our main result establishes an upper bound on the F-free chromatic number of H in terms of the maximum F-free chromatic number of the Gi and the following number-theoretic concept. A set S of natural numbers is k-multiplicative Sidon if ax = by implies a = b and x = y whenever x, y ∈ S and 1 ≤ a, b ≤ k. Suppose that χ(Gi, F) ≤ k and S is a k-multiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k•max S. We then prove that the maximum density of a k-multiplicative Sidon set is Θ(1/ log k). It follows that χ(H, F) ≤ O(dk log k). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature.

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Section: An Improved Constructionmentioning

confidence: 89%

Colourings of the cartesian product of graphs and multiplicative Sidon sets

Pór

Wood

2009

Combinatorica

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“…The sequence (a n ) n \ 1 is the fixed point of the morphism 1 Q 121 and 2 Q 12221 (see for example [4, p. 307] or [55, p. 354]; other occurrences of this sequence can be found in [3,14]). As in the previous example, we deduce that a is a transcendental number (of course a is a homographic image of the number r 0 above).…”

Section: Sequences (1+([na] Mod 2)) N \mentioning

confidence: 99%

Transcendence of Sturmian or Morphic Continued Fractions

Allouche

Davison

Queffélec

et al. 2001

Journal of Number Theory

100

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We prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values, and begins with arbitrary long blocks which are ''almost squares,'' then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence (1+ (NnaM mod 2)) n \ 0 , or are a ''repetitive'' fixed point of a binary morphism satisfying some technical conditions.

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“…(It is easily seen that this leads to the same formula f k = (2 k + (−1) k−1 )/3, although it was not given explicitly in [7].) Set…”

Section: Introductionmentioning

confidence: 90%

“…It should be said that (3) is equivalent to the main result of [7], where the authors used the formula f k+1 = 2f k + (−1) k instead of (4). (It is easily seen that this leads to the same formula f k = (2 k + (−1) k−1 )/3, although it was not given explicitly in [7].)…”

Section: Introductionmentioning

confidence: 97%

On a sequence related to that of Thue–Morse and its applications

Dubickas

2007

Discrete Mathematics

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It is known that the sequence 1, of lengths of blocks of identical symbols in the Thue-Morse sequence has several extremal properties among all non-periodic sequences of the symbols 1 and 2. Its generatingIn terms of combinatorics on words, for any given x ∈ (0, 1) and > 0, we prove that every non-periodic word of an alphabet {1, 2} has a suffix s whose generating function S(x) satisfies the inequality xS(−x) > 1−W (−x)− . Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts { b n }, n = 0, 1, 2, . . ., where b < − 1 is a negative rational number and is a real number. Our results show, for example, that, for any real number = 0, the sequence of fractional parts { (−3/2) n }, n = 0, 1, 2, . . ., has a limit point greater than 0.466452. Furthermore, for each integer b −2 and each real number / ∈ Q, we prove that lim inf n→∞ { b n } ∞ k=1 (1−|b| −(2 k +(−1) k−1 )/3 ) and show that this inequality is sharp.

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A relative of the Thue-Morse sequence

Abstract: We study a sequence, c, which encodes the lengths of blocks in the Thue-Morse sequence. In particular, we show that the generating function for c is a simple product.

Cited by 32 publications

References 3 publications

Colourings of the cartesian product of graphs and multiplicative Sidon sets

Colourings of the cartesian product of graphs and multiplicative Sidon sets

Transcendence of Sturmian or Morphic Continued Fractions

On a sequence related to that of Thue–Morse and its applications

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