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

K. L. Verma

doi:10.68676/saumat.1986491

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

Abstract

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.

Keywords

Generalized third order linear recurrences, Characteristic equation, Explicit formula, Tetradiagonal matrix, Companion matrix

Ethical Statement

This article is a theoretical study in the field of pure mathematics. It does not involve any studies with human participants or animals performed by the author. The author declares that all scientific and ethical principles were strictly followed during the preparation of this manuscript, and all referenced works are properly cited in the bibliography.

Thanks

We sincerely thank the Editor and the reviewers for their valuable comments and constructive suggestions, which have helped us improve the quality and clarity of the manuscript.

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Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

K. L. Verma ^*
0000-0002-6486-8736
India

Early Pub Date

October 19, 2026

Publication Date

-

Submission Date

July 3, 2026

Acceptance Date

October 17, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.68676/saumat.1986491

IZ

https://izlik.org/JA84RZ49DD

APA

Verma, K. L. (2026). Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations. Sakarya Journal of Mathematics, Advanced Online Publication, 84-95. https://doi.org/10.68676/saumat.1986491

AMA

1.Verma KL. Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations. Sakarya Journal of Mathematics. 2026;(Advanced Online Publication):84-95. doi:10.68676/saumat.1986491

Chicago

Verma, K. L. 2026. “Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations”. Sakarya Journal of Mathematics, no. Advanced Online Publication: 84-95. https://doi.org/10.68676/saumat.1986491.

EndNote

Verma KL (October 1, 2026) Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations. Sakarya Journal of Mathematics Advanced Online Publication 84–95.

IEEE

[1]K. L. Verma, “Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations”, Sakarya Journal of Mathematics, no. Advanced Online Publication, pp. 84–95, Oct. 2026, doi: 10.68676/saumat.1986491.

ISNAD

Verma, K. L. “Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations”. Sakarya Journal of Mathematics. Advanced Online Publication (October 1, 2026): 84-95. https://doi.org/10.68676/saumat.1986491.

JAMA

1.Verma KL. Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations. Sakarya Journal of Mathematics. 2026;:84–95.

MLA

Verma, K. L. “Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations”. Sakarya Journal of Mathematics, no. Advanced Online Publication, Oct. 2026, pp. 84-95, doi:10.68676/saumat.1986491.

Vancouver

1.K. L. Verma. Explicit Formula and Matrix Representations of Generalized Third-Order Recurrence Relations. Sakarya Journal of Mathematics. 2026 Oct. 1;(Advanced Online Publication):84-95. doi:10.68676/saumat.1986491