Exact Algebraic Characterization of General Third-Order Linear Recurrence and Identities
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Abstract
This paper presents generalized matrix representations for three distinct Tribonacci-type sequences, constructed by varying their initial conditions alongside a general companion matrix applicable to terms with negative subscripts. These matrices provide a formal framework for studying third-order linear recurrence relations with arbitrary constant coefficients. Using matrix methods and structured algebraic techniques, we derive explicit formulas and new sequence identities, including summation, subtraction, and multiple-argument formulas, as well as product expressions. Furthermore, by systematizing telescoping sums and establishing higher-order variants such as the general Cassini–Simson identity, this work extends classical second-order sequence methodologies to third-order recurrences. The proposed computational framework offers a unified approach for generating and verifying higher-order sequence identities, paving the way for further extensions to Quadranacci and general m-step recursive sequences.
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