On higher order jacobsthal numbers

In this thesis, we obtain various combinatorial formulas and matrix representations of higher order Jacobsthal numbers and their certain special cases. Then in the last chapter, we define generalized third order Jacobsthal matrix sequence and investigate their properties. This thesis consists of seven chapters. The organization of this thesis is as follows: Chapter 1 provides a concise overview of what this thesis is about. Besides, there are properties of generalized second and higher order Fibonacci, Pell and Jacobsthal sequences. The Fibonacci sequence is an important integer sequence with many interesting properties in the literature. This sequence has a number of applications within diverse disciplines. For example, Statistics, Biology, Physics, Finance, Architecture, Computer Science, Medicine are some application areas of this sequence. We begin the first chapter by presenting some properties of the Fibonacci sequence. Then, we present the properties of Pell numbers and some properties of Jacobsthal sequences that appear in many fields such as Fibonacci sequences. Jacobsthal numbers, which are the subject of this thesis, have applications in many branches of mathematics such as group theory, calculus, applied mathematics, linear algebra. Chapter 2 includes some basic definitions and some significant theorems used throughout the thesis. We present some basic definitions, theorems and properties of Jacobsthal and Jacobsthal-Lucas numbers. Chapter 3 presents several combinatorial formulas and matrix representations of the generalized third order Jacobsthal sequences. In the continuation of Chapter 3, we explore in detail four special cases of generalized third order Jacobsthal sequences: third order Jacobsthal sequence, third order Jacobsthal-Lucas sequence, modified third order Jacobsthal sequence and third order Jacobsthal Perrin sequence. Throughout the chapter, some basic properties of generalized third order Jacobsthal numbers such as Binet formulas, generating functions, and Simson formulas are presented. Chapter 4 includes the six special cases and some properties of generalized fourth order Jacobsthal sequences. In the continuation of Chapter 4, we investigate, in detail, six special cases: fourth order Jacobsthal sequence, fourth order Jacobsthal-Lucas sequence, modified fourth order Jacobsthal sequence, fourth order Jacobsthal Perrin sequence, adjusted fourth order Jacobsthal sequence and modified fourth order Jacobsthal-Lucas sequence. Chapter 5 consists of the basic information about generalized fifth order Jacobsthal numbers. Firstly, in this chapter, we present the definition and some properties of generalized Pentanacci sequence. In the following, we give six special cases of the fifth order Jacobsthal numbers and their properties. Chapter 6 includes the some properties of the generalized sixth order Jacobsthal sequences and its six special cases. First we present the definition and some properties of a generalized Hexanacci sequence. After that, we give six special cases of the sixth order Jacobsthal numbers and their Binet formulas, generating functions, Simson formulas, etc. Chapter 7 includes matrix sequences of generalized third order Jacobsthal numbers and to examine the relationship between generalized third order Jacobsthal matrix sequences and their special cases. In addition, in this chapter, we give the n-th (general) term of the generalized third order Jacobsthal matrix sequence.