Locating terms in the Stern–Brocot tree

Bates, Bruce; Bunder, Martin W.; Tognetti, Keith

doi:10.1016/j.ejc.2007.10.005

Cited by 18 publications

(22 citation statements)

References 5 publications

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“…Example 43. For m = 3, the sequence counting admissible subwords associated with the Tribonacci numeration system starts with 1,2,3,3,4,5,5,5,7,8,6,7,7,6,9,11,9,11,12,10,9,11,11,9,7,11,14,12,15,17,15,14,18,19,15,14,14,11,15,17,15,15,17,15,8,13,17,15,19,22, . .…”

Section: Discussionmentioning

confidence: 99%

“…Plugging in the first few terms of (S(n)) n≥0 in Sloane's On-Line Encyclopedia of Integer Sequences [33], it seems to be a shifted version of the sequence A007306 of the denominators occurring in the Farey tree (left subtree of the full Stern-Brocot tree), which contains every (reduced) positive rational less than 1 exactly once. Many papers deal with this tree; for instance, see [4,18]. The Farey tree is an infinite binary tree made up of mediants.…”

Section: The Sequence A007306 Farey Tree and Stern-brocot Sequencementioning

confidence: 99%

“…Let (S F (n)) n≥0 be the sequence whose nth term, n ≥ 0, is the number of non-zero elements in the nth row of P F . Hence, the first few terms of this sequence are 1,2,3,4,4,5,6,6,6,8,9,8,8,7,10,12,12,12,10,12,12,8,12,15,16,16,15, . .…”

Section: Generalizations: the Fibonacci Casementioning

confidence: 99%

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Counting the number of non-zero coefficients in rows of generalized Pascal triangles

Leroy

Rigo

Stipulanti

2017

Discrete Mathematics

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Abstract. This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence (S(n)) n≥0 counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for (S(n)) n≥0 . This leads to a connection with the 2-regular Stern-Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F -regularity of the corresponding sequence.

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Section: Discussionmentioning

confidence: 99%

Section: The Sequence A007306 Farey Tree and Stern-brocot Sequencementioning

confidence: 99%

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Counting the number of non-zero coefficients in rows of generalized Pascal triangles

Leroy

Rigo

Stipulanti

2017

Discrete Mathematics

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“…Some of the recent developments include the following. In [1], B. Bates et al gave a simple method of identifying both the level and the position within the level of each fraction in the Stern-Brocot tree. In [4], K. Dilcher and K. B. Stolarsky discovered an interesting polynomial analogue to the Stern sequence.…”

Section: Introductionmentioning

confidence: 99%

Analogs of the Stern Sequence

Chan¹

2011

Integers

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“…Bates et al [5] have shown that when expressed in the shortest form of their simple continued fractions: i) Left mediants possess continued fractions of the form [a 0 , a 1 , . .…”

Section: Definition 11 (Left and Right Mediants)mentioning

confidence: 99%

Mirroring and interleaving in the paperfolding sequence

Bates

Bunder

Tognetti

2010

APPL ANAL DISCRETE M

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Three equivalent methods of generating the paperfolding sequence are presented as well as a categorisation of runs of identical terms. We find all repeated subsequences, the largest repeated subsequences and the spacing of singles, doubles and triples throughout the sequence. The paperfolding sequence is shown to have links to the Binary Reflected Gray Code and the Stern-Brocot tree. Three equivalent methods of generating the paperfolding sequence are presented as well as a categorisation of runs of identical terms. We find all repeated subsequences, the largest repeated subsequences and the spacing of singles, doubles and triples throughout the sequence. The paperfolding sequence is shown to have links to the Binary Reflected Gray Code and the Stern-Brocot tree. Disciplines Physical Sciences and Mathematics

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Locating terms in the Stern–Brocot tree

Cited by 18 publications

References 5 publications

Counting the number of non-zero coefficients in rows of generalized Pascal triangles

Counting the number of non-zero coefficients in rows of generalized Pascal triangles

Analogs of the Stern Sequence

Mirroring and interleaving in the paperfolding sequence

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