On the Padovan p-numbers

Abstract

In this paper, we define the Padovan p-numbers and then we obtain their miscellaneous properties such as the generating matrix, the Binetformula, the generating function, the exponential representation, the combinatorial representations, the sums and permanental representation. Also, we study the Padovan p-numbers modulo m. Furthermore, we define Padovan p-orbit of a finite group and then, we obtain the length of the Padovan p-orbits of the quaternion group $Q_{2^n}$, $(n\geq 3)$.