Matrix Representation of Generalized Quadranacci Number Sequences

Abstract

In this paper, we present a matrix representation for the generalized Quadranacci number sequence, defined by the 4^th-order generalized recurrence relation, initial terms , and constant coefficients , The parameters , and  are arbitrarily chosen real numbers. Fundamental results based on this definition are established in general symbolic form. A generalized companion matrix, associated with the recurrence relation, is introduced to analyze the properties of Quadranacci numbers. Subsequently, is derived in a generalized form, enabling the application of matrix techniques to study the properties of Quadranacci number sequences. By appropriately specifying the initial values, and the constant coefficients,, several existing results are shown to be special cases of the derived results. Moreover, all the results obtained are implicitly applicable to the generalized Tribonacci and Fibonacci sequences, which are governed by lower-order recurrence relations.