Communications of the
Korean Mathematical Society CKMS

The Padovan sequence is the third-order linear recurrence $(\mathcal{P}_n)_{n\geq 0}$ defined by $\mathcal{P}_n=\mathcal{P}_{n-2}+\mathcal{P}_{n-3}$ for all $n\geq 3$ with initial conditions $\mathcal{P}_0=0$ and $\mathcal{P}_1=\mathcal{P}_2=1$. In this paper, we investigate a generalization of the Padovan sequence called the $k$-generalized Padovan sequence which is generated by a linear recurrence sequence of order $k\geq 3$. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences.

Keywords: Fibonacci number, generalized Padovan number, recurrence sequence

MSC numbers: 11B37, 11B39

Supported by: The first author was supported in part by Project VRI ID 5385 (Universidad del Cauca). The second author thanks the Universidad del Cauca for support during his Ph.D. studies.