Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations

This paper investigates third-order linear recurrence relations and derives an explicit formula for generalized tribonacci sequences in terms of the roots of their characteristic equations. Using these roots, we establish connections between generalized tribonacci sequences and matrix theory by introducing novel tetradiagonal and companion matrices. In particular, we prove that the n-th term of the sequence is identically equal to both the determinant of the associated tetradiagonal matrix and the first entry of the n-th power of the companion matrix. These determinant representations are further employed to derive several algebraic identities for the generalized tribonacci sequence, including a Vajda-type identity and related identities involving shifted terms. Finally, the resulting determinant and sequence identities are expressed explicitly through a double-summation formula, providing an alternative combinatorial representation of the general term.