Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices
Abstract
We study matrix representations of the Jacobsthal sequence generated by binary 3x3 matrices with determinants 0, 2, and -2, using linear algebraic methods analogous to Fibonacci-type constructions. Explicit formulas for matrix powers are obtained, yielding identities for Jacobsthal numbers, including convolution formulas, trace relations, determinant expressions, and Cassini-type identities. We further derive congruence relations and recurrence formulas, and analyze arithmetic properties such as partial sums and the Sidon-type structure of the sequence. Finally, we prove that exactly three conjugacy classes of binary 3x3 matrices generate the Jacobsthal sequence, providing a unified algebraic framework that connects matrix theory with second-order linear recurrences.
Related
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BibTeX
@misc{martinez2026matrix,
title = {{Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices}},
author = {Wilson Arley Martinez and Samin Ingrith Ceron},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30069},
eprint = {2608.30069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30069},
}
RIS
TY - UNPB
TI - Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices
AU - Wilson Arley Martinez
AU - Samin Ingrith Ceron
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30069
UR - https://arxiv.org/abs/2608.30069
AB - We study matrix representations of the Jacobsthal sequence generated by binary 3x3 matrices with determinants 0, 2, and -2, using linear algebraic methods analogous to Fibonacci-type constructions. Explicit formulas for matrix powers are obtained, yielding identities for Jacobsthal numbers, including convolution formulas, trace relations, determinant expressions, and Cassini-type identities. We further derive congruence relations and recurrence formulas, and analyze arithmetic properties such as partial sums and the Sidon-type structure of the sequence. Finally, we prove that exactly three conjugacy classes of binary 3x3 matrices generate the Jacobsthal sequence, providing a unified algebraic framework that connects matrix theory with second-order linear recurrences.
ER -