In this paper, we provide properties and applications of some special integer sequences obtained from a difference equation of degree k. These sequences are used to define some special elements in algebras obtained by the Cayley-Dickson process. For example, the association between quaternions and Fibonacci numbers leds to the appearance of new special elements whose properties give us new applications: in Coding Theory or in finding new properties for the algebras on they were defined, etc. Moreover, some properties of these elements are also presented. Were described the circumstances in which some algebraic structures on some sets defined by these numbers can be obtained and are provided conditions such that these elements are invertible.