Initial numbers for Lucas? type series have so far been established only Fibonacci (2,1) and Tribonacci (3,1,3) sequences. Characteristics of stated is their asymptotic relation with the exponent constant. By using a simple procedure based on relations exponents sequences constant application Nearest Integer Function - NIF, general rule initial Generalized sequence has established, first time. ...

The Horadam sequence is a generalization of the Fibonacci numbers in the complex plane, depending on a family of four complex parameters: two recurrence coefficients and two initial conditions. The necessary and sufficient periodicity conditions formulated in [1] are used to enumerate all Horadam sequences with a given period [2]. The geometry of periodic orbits is analyzed, where regular star-...

In this paper, we determine the maximum number of distinct Lyndon factors that a word of length n can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length n on an alphabet of size σ, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length n is 1 and the minimum tot...

[n is an arbitrary natural number, r is an arbitrary (nonzero) real quantity) gives & positive integer k. Since both r and k turn out to be Fibonacci number ratios, the results established in this paper can be viewed as a particular kind of Fibonacci identities that are believed to be new [see (4.7) and (4.8)]. Throughout the paper we shall make use of the following properties of the Fibonacci ...

Fibonacci trees and generalized Fibonacci trees have been defined and studied by Horibe [4], Chang [2], and the author [1]. The k^ tree in the sequence of r-ary generalized Fibonacci trees, T(k), has T(k-c(i)) as the I leftmost subtree descending from its root node for k>r, and T(k) consists of a single root node for k-\ ...,r. Here c(i), z' = l,...,r, are positive integers with greatest common...

The Binet formula gives a natural way for Fibonacci numbers to be viewed as a function of a complex variable. We experimentally study the complex dynamics of the Fibonacci numbers viewed in that manner. Attracting and repelling fixed points are related to the filled Julia set and to regions of escape time images with fascinating behavior. Introduction The Fibonacci numbers are traditionally des...

Let Fn be the nth Fibonacci number. For 1 ≤ k ≤ m − 1 let

Here, we prove that there is no perfect number of the form Fmn/Fm, where Fk is the kth Fibonacci number.

for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure 1.1. The points in R2 associated with ( i j ) , ( i+1 j ) , and ( i+1 j+1 ) form a unit equilateral triangle. This arrayal is called the natural arrayal of Pascal’s triangle in R2. For all t ∈ R : −√3 < t < √3 and nonnegative integers k, define k(t) to be the sum of all binomial coefficients associated with points in R2...

The design of Interleaver/Deinterleavers using Fibonacciclass quasistructures is proposed. We introduce an optical passive configuration composed of Fibonacci quasistructures and circulators which acts as interleaver and deinterleaver. Odd and even channels are interleaved/deinterleaved with dense wavelength-division multiplexing (DWDM) multichannel filter based on Fibonacci quasi-periodic stru...