Matrix Representations Of Jacobsthal Numbers And Applications

In combinatorial mathematics, there are many sequences, such as Fibonacci se-quences, Jacobsthal sequences, Lucas sequences, Pell sequences, Catalan sequences,binomial coefcient, etc. These sequences have deep combinatorial background, sowe call them combinatorial sequences.Combinatorial sequences come from counting problems and they are used ascounting tools. It is signifcant to investigate combinatorial sequences. In this paper,using the matrix methods, we study some properties of Jacobsthal numbers and theirgeneralizations.In Chapter1, we give some general counting models in enumerative combi-natorics. The recursive relations, generating functions, explicit expressions, andcombinatorial meaning of Jacobsthal numbers are introduced.In Chapter2, we present some properties of Jacobsthal-Lucas numbers. We alsoexplain how Koken, Bozkurt investigated the relations between Jacobsthal sequencesand the related Jacobsthal-Lucas sequences using matrix methods.In Chapter3, we obtain two important properties by giving a new initial con-dition to some generalized Jacobsthal-Lucas numbers of order κ.In Chapter4, some generalized Fibonacci-Jacobsthal sequences are introducedand studied by using matrix methods.In Chapter5, we defne a new family of κ-Jacobsthal sequences by using theBinet formula for Jacobsthal numbers. We also discuss some properties of thesesequences.