In this paper, we define the complex-type Padovan-p sequence and then give the relationships between the Padovan-p numbers and the complex-type Padovan-p numbers. Also, we provide a new Binet formula and a new combinatorial representation of the complex-type Padovan-p numbers by the aid of the nth power of the generating matrix of the complex-type Padovan-p sequence. In addition, we derive various properties of the complex-type Padovan-p numbers such as the permanental, determinantal and exponential representations and the finite sums by matrix methods.

The arrowhead-Pell-random-type sequence was defined by Erdag et al. [5]. In this work, we redefine by the means of the elements of the groups. Then we examine the sequence in finite groups in detail. Also, we obtain the lengths of the periods of the generalized Quaternion group Q_2m as applications of the results obtained.