Some properties of Padovan matrices and bi-periodic Padovan matrices

Let $\left(P_{n}\right){n\geq0}$ be the sequence of bi-periodic Padovan numbers and let $\left(M{p_{n}}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan matrices. In this article we study when these matrices are diagonalizable and we obtain a certain connection with the Lucas number sequence. We also obtain some connections of these matrices with the generating matrix $Q$ for the Padovan numbers.

💡 Research Summary

The paper investigates the algebraic properties of Padovan matrices and their bi‑periodic generalizations, focusing on diagonalizability, commutativity, and connections with classical Lucas numbers.

The classical Padovan sequence (p_n) is defined by the third‑order recurrence (p_n=p_{n-2}+p_{n-3}) with initial values (p_0=p_1=p_2=1). Its generating matrix
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