通过二元3×3矩阵研究雅可比施塔尔数的矩阵表示与算术性质
Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices
- Universidad del Cauca（考卡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过二元3×3矩阵的线性代数方法，推导雅可比施塔尔数的矩阵幂公式与多种恒等式，分析其算术性质，确定生成该序列的共轭类数量，建立矩阵理论与二阶线性递推的统一框架。
AI中文摘要：
我们采用类似斐波那契型构造的线性代数方法，研究由行列式为0、2和-2的二元3×3矩阵生成的雅可比施塔尔序列的矩阵表示，得到矩阵幂的显式公式，推导出雅可比施塔尔数的恒等式，包括卷积公式、迹关系、行列式表达式及卡西尼型恒等式，进一步推导同余关系与递推公式，分析部分和、序列的西顿型结构等算术性质，最终证明恰好有三类二元3×3矩阵的共轭类可生成雅可比施塔尔序列，提供了连接矩阵理论与二阶线性递推的统一代数框架。
英文摘要：
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.
