Among numerical sequences, the Fibonacci numbers have achieved a kind of celebrity status with such fabulous properties, it is no wonder that the Fibonacci numbers stand out as a kind of super sequence. Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is consequently, incredibly vast. The Fibonacci sequence has been generalized in a num...

In 1982, Prodinger and Tichy defined the Fibonacci number of a graph G to be the number of independent sets of the graph G. They did so since the Fibonacci number of the path graph Pn is the Fibonacci number Fn+2 and the Fibonacci number of the cycle graph Cn is the Lucas number Ln. The tadpole graph Tn,k is the graph created by concatenating Cn and Pk with an edge from any vertex of Cn to a pe...

A frame is a square uu, where u is an unbordered word. Let F (n) denote the maximum number of distinct frames in a binary word of length n. We count this number for small values of n and show that F (n) is at most bn/2c + 8 for all n and greater than 7n/30− for any positive and infinitely many n. We also show that Fibonacci words, which are known to contain plenty of distinct squares, have only...

A Fibonacci poem follows the sequence to any length in its count of syllables per line, or words lines stanza, other countable thing connected with poem. Fib is a special case poem, 6 whose syllable line first numbers sequence: 1; 2; 3; 5; 8. The present collection Fibs arose out Bridges poetry community response organization’s efforts make 2021 conference more interactive.

The notion of the Fibonacci cobweb poset from [1] has been naturally extended to any admissible sequence F in [2] where it was also recognized that the celebrated prefab notion of Bender and Goldman [3] (see also [4,5]) admits such an extension so as to encompass the new type combinatorial objects from [2] as leading examples. Recently the present author had introduced also [6] two natural part...

A Fibonacci integer is an integer in the multiplicative group generated by the Fibonacci numbers. For example, 77 = 21 · 55/(3 · 5) is a Fibonacci integer. Using some results about the structure of this multiplicative group, we determine a near-asymptotic formula for the counting function of the Fibonacci integers, showing that up to x the number of them is between exp(c(log x) − (log x)) and e...

Smooth infinite words over Σ = {1, 2} are connected to the Kolakoski word K = 221121 · · ·, defined as the fixpoint of the function ∆ that counts the length of the runs of 1’s and 2’s. In this paper we extend the notion of smooth words to arbitrary alphabets and study some of their combinatorial properties. Using the run-length encoding ∆, every word is represented by a word obtained from the i...

Denote by sq(w) the number of distinct squares in a string w and let S be the class of standard Sturmian words. They are generalizations of Fibonacci words and are important in combinatorics on words. For Fibonacci words the asymptotic behaviour of the number of runs and the number of squares is the same. We show that for Sturmian words the situation is quite different. The tight bound 8 10 |w|...

It is proved that the asymptotic average eccentricity and the asymptotic average degree of both Fibonacci cubes and Lucas cubes are (5 + √ 5)/10 and (5 − √ 5)/5, respectively. A new labeling of the leaves of Fibonacci trees is introduced and it is proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. ...