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.

In this article, we define the complex-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the complex-type k-Fibonacci numbers. Also, we obtain miscellaneous properties of the complex-type k-Fibonacci numbers such as the Binet formulas, the combinatorial, permanental, determinantal representations and the sums. In addition, we study the complex-type k-Fibonacci sequence modulo m and then we give some results concerning the periods and the ranks of the complex-type k-Fibonacci sequences for any k and m which are related the periods of the k-step Fibonacci sequences modulo m. Furthermore, we extend the complex-type k-Fibonacci sequences to groups. Finally, we obtain the periods of the complex-type 2-Fibonacci sequences in the dihedral group $D_{2m},(m\ge 2)$ with respect to the generating pairs (x, y) and (y, x).

This note considers some real and complex extensions and generalizations of the Leonardo sequence, which is embedded within each of these two types of intriguing sequences, intriguing because there are still some unanswered questions. The connections between inhomogeneous and homogeneous forms are used as examples of a possible reason that the Leonardo sequences have been, in a sense, historically neglected.

Recursive sequences with gaps have been studied previously. This paper considers some elementary properties of such sequences where the gaps have been created on a regular basis from sequence to sequence – ‘discatenated’ (systematic gaps) and ‘lacunary’ (general gaps). In particular, their generating functions are developed in order to open up their general terms and relations with other properties.