The quaternion-Pell sequence

In this paper, we define the quaternion-Pell sequence and derive the generating matrix of the sequence. Then we produce the semigroups using the multiplicative orders of the generating matrix of the quaternion-Pell sequence when read modulo α. We also study the quaternion-Pell sequence modulo $\alpha ,$ and then we give the relationship among the periods of the quaternion-Pell sequence modulo α and the orders of the semigroups obtained. In addition, we extend the quaternion-Pell sequence to groups. Finally, we obtain the periods of the quaternion-Pell sequences in dihedral groups $D_{2m}$ as applications of the results obtained.