Browsing by Subject "Fibonacci numbers"

Now showing 1 - 3 of 3

Open Access
Aspects of Fibonacci numbers
(1994) Yücel, Gülnihal
This thesis consists of two parts. The first part, which is Chapter 2, is a survey on some aspects of Fibonacci numbers. In this part, we tried to gather some interesting properties of these numbers and some topics related to the Fibonacci sequence from various references, so that the reader may get an overview of the subject. After giving the basic concepts about the Fibonacci numbers, their arithmetical properties are studied. These include divisibility and periodicity properties, the Zeckendorf Theorem, Fibonacci trees and their relations to the representations of integers, polynomials used for deriving new identities for Fibonacci numbers and Fibonacci groups. Also in Chapter 2, natural phenomena related to the golden section, such as certain plants having Fibonacci numbers for the number of petals, or the relations of generations of bees with the Fibonacci numbers are recounted. In the second part of the thesis. Chapter 3, we focused on a Fibonacci based random number sequence. We analyzed and criticized the generator Sfc = k(j>—[k(j)] by applying some standart tests for randomness on it. Chapter 5, the Appendix consists of Fortran programs used for executing the tests of Chapter 3.
Open Access
On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices
(Elsevier Inc., 2016) Altinişik, E.; Keskin, A.; Yildiz, M.; Demirbüken, M.
Let Kn be the set of all n×n lower triangular (0,1)-matrices with each diagonal element equal to 1, Ln={YYT:Y ∈ Kn} and let cn be the minimum of the smallest eigenvalue of YYT as Y goes through Kn. The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that cn is equal to the smallest eigenvalue of Y0Y0 T, where Y0 ∈ Kn with (Y0)ij = (Formula presented.) for i > j. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices. © 2015 Elsevier Inc.
Open Access
Pattern-avoiding permutations: The case of length three, four, and five
(2023-06) Akbaş, Zilan
A shorter permutation of length k is said to appear as a pattern in a longer per-mutation of length n if the longer permutation has a subsequence of length k that is order isomorphic to the shorter one. Otherwise, the longer permutation avoids the shorter one as a pattern. We use Sn(τ) to denote the set of permutations of length n that avoid pattern τ. Pattern avoidance induces an equivalence relation on the pattern set Sk. For ρ, τ ∈ Sk, we define the equivalence relation as follows: ρ ∼W τ if and only if |Sn(ρ)| = |Sn(τ)| for all n ≥ 1. The equivalence classes of this relation are called Wilf classes. The main questions are determining the Wilf classes of Sk and enumerating each class. We first study the Wilf classification and enumeration of each class for S3 and S4. We then present some new numerical results regarding the Wilf classification of pairs of patterns of length five. We define a Wilf class as small if it contains only one pair and big if it contains more than one pair. We show that there are at least 968 small Wilf classes and at most 13 big Wilf classes.