Generalized Jacobsthal numbers

In this study, by considering general properties of Jacobsthal and Jacobsthal-Lucas numbers, definitions and properties of order-k Jacobsthal and order-k Jacobsthal-Lucas sequences, which are the generalizations of Jacobsthal and Jacobsthal-Lucas sequences, are given. In the first chapter, history of Fibonacci and Lucas sequences that is considered as foundation of integer sequences and relations between these numbers and matrices are mentioned. In the second chapter, after definitions of some integer sequences which are important in the literature are given, some fundamental concepts of these sequences are stated. In addition to properties of Jacobsthal and Jacobsthal-Lucas numbers, summation properties and matrix representation are obtained in the third chapter. In the last chapter, in addition to definitions, generating matrices, eigenvalues and Binet formulas of generalized order-k Jacobsthal numbers and Jacobsthal k-numbers, their generating functions and combinatorial representations are obtained.