The random Fibonacci sequence is defined by t1 = t2 = 1 and tn = ±tn−1 + tn−2, for n > 3, where each ± sign is chosen at random with probability P (+) = P (−) = 12 . Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate 1.13198824 . . . . We will consider what happens to random Fibonacci sequences when we remove the randomness; specifically, we will choos...

We enumerate and characterize some classes of alternating reverse involutions avoiding a single pattern length three or four. If on one hand the case patterns is trivial, other hand, four more challenging involves sequences combinatorial interest, such as Motzkin Fibonacci numbers.

Variable-length splittable codes are derived from encoding sequences of ordered integer pairs, where one of the pair’s components is upper bounded by some constant, and the other one is any positive integer. Each pair is encoded by the concatenation of two fixed independent prefix encoding functions applied to the corresponding components of a pair. The codeword of such a sequence of pairs cons...

The combinatorial properties of the Fibonacci innnite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity , quasicrystals etc. In this note, we introduce the singular words of the Fibonacci innnite word and discuss their properties. We establish two decompositions of the Fibonacci word in singu...

It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound ⌈log 2 (n)⌉ + 1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is ⌈logφ(n)⌉ + 1, where φ denotes the gol...

We look at binary strings of length n which contain no odd run of zeros and express the total number of such strings, the number of zeros, the number of ones, the total number of runs, and the number of levels, rises and drops as functions of the Fibonacci and Lucas numbers and also give their generating functions. Furthermore, we look at the decimal value of the sum of all binary strings of le...

A Fibonacci string is a length n binary string containing no two consecutive 1s. Fibonacci cubes (FC), Extended Fibonacci cubes (ELC) and Lucas cubes (LC) are subgraphs of hypercube defined in terms of Fibonacci strings. All these cubes were introduced in the last ten years as models for interconnection networks and shown that their network topology posseses many interesting properties that are...

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79–95, 2011) proved that at every position of an infinite Sturmian word starts an abelian power of exponent k, for every positive integer k. Here, we improve on this result, studying the maximal exponent of abelian powers and abelian repetitions (an abelian repetition is the analogous of a fractional power in the abelian setting) occurring in...

On the example of the Fibonacci chain we show how to efˇciently generate inˇnite words deˇned by substitutions. We introduce substitution trees and we present an algorithm that uses them to generate such inˇnite words. We use a stack to generate the words in a completely different manner than in traditional breadth-ˇrst tree traversals. We show that the algorithm has O(logn) space complexity an...

Knödel graphs and Fibonacci graphs are two classes of bipartite incident-graph of circulant digraphs. Both graphs have been extensively studied for the purpose of fast communications in networks, and they have deserved a lot of attention in this context. In this paper, we show that there exists an O(n log n)-time algorithm to recognize Knödel graphs, and that the same technique applies to Fibon...