# Combinatorial Properties of Fibonacci Arrays.

13 Apr 2019-pp 448-466

TL;DR: The set of all Fibonacci arrays is a 2D primitive language (under certain conditions), count the number of borders in Fib onacci arrays, and show that the set ofall Fiboncians is a non-recognizable language.

Abstract: The non-trivial extension of Fibonacci words to Fibonacci arrays was proposed by Apostolico and Brimkov in order to study repetitions in arrays. In this paper we investigate several combinatorial as well as formal language theoretic properties of Fibonacci arrays. In particular, we show that the set of all Fibonacci arrays is a 2D primitive language (under certain conditions), count the number of borders in Fibonacci arrays, and show that the set of all Fibonacci arrays is a non-recognizable language. We also show that the set of all square Fibonacci arrays is a two dimensional code.

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TL;DR: In this paper , the exact number of tandems in a given finite Fibonacci array fm,n was shown to be O(m2nlogn) for any given m,n ≥ 1 using a two-dimensional homomorphism.

2 citations

TL;DR: In this paper , the subwords of the two-dimensional inﬁnite Fibonacci word f ∞ , ∞ were enumerated in a few possible ways.

Abstract: Given an inﬁnite word, enumerating its subwords is an important exercise for understanding the structure of the word. The process of ﬁnding all the subwords is quite tricky for two-dimensional words. In this paper we enumerate the subwords of the two-dimensional inﬁnite Fibonacci word, f ∞ , ∞ , in a few possible ways. In addition, we extend a method for locating the subwords of the one-dimensional inﬁnite Fibonacci word f ∞ to locate the positions of the subwords of f ∞ , ∞ .

09 Jul 2022

TL;DR: In this article , two possible ways of enumerating the factors of the fixed point of the sequence of Fibonacci arrays and a method for locating these factors in $f{ ∞, ∞}$ are explored.

Abstract: Given an infinite word, enumerating its factors is an important exercise for understanding the structure of the word. The process of finding all the factors is quite tricky for two-dimensional words. In this paper, two possible ways of enumerating the factors of the fixed point ($f_{\infty,\infty}$) of the sequence of Fibonacci arrays and a method for locating these factors in $f_{\infty,\infty}$ are explored. In addition, the factor complexity and the locations of the factors of the fixed point of Fibonacci sequence of arrays are also analysed.

##### References

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01 Jan 1979

TL;DR: This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity, appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.

Abstract: This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity. The authors present the theory in a concise and straightforward manner, with an eye out for the practical applications. Exercises at the end of each chapter, including some that have been solved, help readers confirm and enhance their understanding of the material. This book is appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.

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TL;DR: In this article, Berstel and Perrin proposed the concept of Sturmian words and the plactic monoid, which is a set of permutations and infinite words.

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TL;DR: The aim of this chapter is to generalize concepts and techniques of formal language theory to two dimensions.

Abstract: The aim of this chapter is to generalize concepts and techniques of formal language theory to two dimensions. Informally, a two-dimensional string is called a picture and is defined as a rectangular array of symbols taken from a finite alphabet. A two-dimensional language (or picture language) is a set of pictures.

439 citations

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TL;DR: A two-dimensional generalization of Sturmian sequences corresponding to an approximation of a plane is studied, which deduces a new way of computing the rectangle complexity function and provides an upper bound on the number of frequencies of rectangular factors of given size.

109 citations

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01 Jan 2007

TL;DR: A history and introduction to the Fibonacci Numbers can be found in this paper, along with a survey of the application of the number in art and architecture, as well as the famous Binet Formula for finding a Particular FPN.

Abstract: A History and Introduction to the Fibonacci Numbers The Fibonacci Numbers in Nature The Fibonacci Numbers and the Pascal Triangle The Fibonacci Numbers and the Golden Ratio The Fibonacci Numbers and Continued Fractions A Potpourri of Fibonacci Number Applications The Fibonacci Numbers Found in Art and Architecture The Fibonacci Numbers and Musical Form The Famous Binet Formula for Finding a Particular Fibonacci Number The Fibonacci Numbers and Fractals Epilogue Index.

100 citations