Abstract
In this paper we present applications of special numbers obtained from a difference equation of degree three. As a particular case of this difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these quaternion elements, we can define a set with an interesting algebraic structure, namely, an order on a generalized rational quaternion algebra. Another presented application is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes).
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