Interspersions and dispersions

Kimberling, Clark

doi:10.1090/s0002-9939-1993-1111434-0

Cited by 15 publications

(10 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The number of letters produced by the two substitutions up to the l-th occurence of v is thus 2l + i l − l = i l + l = j l . The position of the first u is 1 = u(1), and since (i l , j l ) form complementary sequences, we obtain: i l = u(l), and by (3),…”

Section: Fibonacci Words Satisfy the Fibonacci Recursionmentioning

confidence: 94%

See 1 more Smart Citation

Labeled Fibonacci Trees

Legendre

2014

Preprint

View full text Add to dashboard Cite

The study describes a class of integer labelings of the Fibonacci tree, the tree of descent introduced by Fibonacci. In these labelings, Fibonacci sequences appear along ascending branches of the tree, and it is shown that the labels at any level are consecutive integers. The set of labeled trees is a commutative group isomorphic to Z 2 , and is endowed with an order relation. Properties of the Wythoff array are recovered as a special instance, and further properties of the labeled Fibonacci trees are described. These trees can be viewed as generalizations of the Wythoff array.

show abstract

Section: Fibonacci Words Satisfy the Fibonacci Recursionmentioning

confidence: 94%

“…Structural properties of the Wythoff array [3] can be read from its representation as the Hofstadter tree.…”

Section: The Hofstadter Treementioning

confidence: 99%

Labeled Fibonacci Trees

Legendre

2014

Preprint

View full text Add to dashboard Cite

show abstract

“…Many variations, interspersions and dispersions have since been given, see e.g., [16], [19]. All are doubly infinite, lim j→∞ (A(i, j + 1) − A(i, j)) = ∞ for every i ≥ 1, and every positive integer appears precisely once in A. .…”

Section: Infinite Complementary Arraysmentioning

confidence: 99%

Complementary Iterated Floor Words and the Flora Game

Fraenkel

2010

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Let ϕ = (1 + √ 5)/2 denote the golden section. We investigate relationships between unbounded iterations of the floor function applied to various combinations of ϕ and ϕ 2 . We use them to formulate an algebraic polynomial-time winning strategy for a new 4-pile take-away game Flora, which is motivated by partitioning the set of games into subsets CompGames and PrimGames. We present recursive, arithmetic and word-mapping winning strategies for it. The arithmetic one is based on the Fibonacci numeration system. We further show how to generate the floor words induced by the iterations using word-mappings and and characterize them using the Fibonacci numeration system. We also exhibit an infinite array of such sequences.

show abstract

“…9 has many wonderful properties, some of which are mentioned here. I learned about most of these properties from John Conway [13], but this array has a long history -see Fraenkel and Kimberling [28], Kimberling [43], [44], [45], [46], [47], [48], [49], Morrison [74] and Stolarsky [96], [97]. It is related to a large number of sequences in the database (the main entry is A35513).…”

Section: The Wythoff Arraymentioning

confidence: 99%