The Fibonacci sequence origin is attributed and referred to the first edition (lost) of “Liber abaci” (1202) by Leonardo Fibonacci [Pisano] (see second edition from 1228 reproduced as Il Liber Abaci di Leonardo Pisano publicato secondo la lezione Codice Maglibeciano by Baldassarre Boncompagni in Scritti di Leonardo Pisano vol. 1, (1857) Rome). Very recently [1, 2] Fibonomial coefficients [–5] h...

For an optimal binary search tree T with a subtree S(d) at a distance d from the root of T, we study the ratio of the weight of S(d) to the weight of T. The maximum possible value, which we call ρ(d), of the ratio of weights, is found to have an upper bound of 2/Fd+3 where Fi is the ith Fibonacci number. For d = 1, 2, 3, and 4, the bound is shown to be tight. For larger d, the Fibonacci bound g...

Five new classes of Fibonacci-Hessenberg matrices are introduced. Further, we introduce the notion of two-dimensional Fibonacci arrays and show that three classes of previously known Fibonacci-Hessenberg matrices and their generalizations satisfy this property. Simple systems of linear equations are given whose solutions are Fibonacci fractions.

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...

As VLSI technology has scale down to the deep sub-micrometer (DSM) technology the global interconnect delay is becoming a large fraction of the overall delay of a circuit. Additionally, the increasing cross-coupling capacitances between wires on the same metal layer create a situation where the delay of a wire is strongly dependent on the electrical state of its neighboring wires. So the crosst...

We study the suitability of the additive lagged-Fibonacci pseudorandom number generator for parallel computation. This generator has a relatively short period with respect to the size of its seed. However, the short period is more than made up for with the huge number of full-period cycles it contains. We call these diierent full-period cycles equivalence classes. We show how to enumerate the e...

where 0) = %(1 + V5) . It is well known that Z(OJ) is a Euclidean domain [6, pp. 214-15], and that the units of Z(oo) are given by ±0), where nEZ [6, p. 221]. The Binet formula _ _ Fn = (00 03)/((A) W) = (0D 0))/>/5, where 0) = %(1 v5) is the conjugate of 0), expresses the n Fibonacci number in terms of the unit 0). Simiarly, the n Lucas number is given by Ln = b) + 0)". Also, an elementary ind...

The Fibonacci dimension fdim(G) of a graph G was introduced in [1] as the smallest integer d such that G admits an isometric embedding into Γd, the d-dimensional Fibonacci cube. The Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacon...

We study the properties of the function R(n) which determines the number of representations of an integer n as a sum of distinct Fibonacci numbers Fk. We determine the maximum and mean values of R(n) for Fk ≤ n < Fk+1. Mathematics Subject Classification. 11A67, 11B39.

As in [1, 2], for rapid numerical calculations of identities pertaining to Lucas or both Fibonacci and Lucas numbers we present each identity as a binomial sum. 1. Preliminaries The two most well-known linear homogeneous recurrence relations of order two with constant coefficients are those that define Fibonacci and Lucas numbers (or Fibonacci and Lucas sequences). They are defined recursively ...