The Jacobsthal sequences in finite groups

‎In this paper‎, ‎we study the generalized order-$k$ Jacobsthal sequences‎ ‎modulo $m$ for $k\geq 2$ and the generalized order\textit{-k‎ }% ‎Jacobsthal-Padovan sequence modulo $m$ for $k\geq 3$‎. ‎Also‎, ‎we define the‎ ‎generalized order-\textit{k} Jacobsthal orbit of a \textit{k}-generator‎ ‎group for $k\geq 2$ and the generalized order-\textit{k } Jacobsthal-Padovan‎ ‎orbit a \textit{k}-generator group for $k\geq 3$‎. ‎Furthermore‎, ‎we obtain the‎ ‎lengths of the periods of the generalized order-3 Jacobsthal orbit‎ ‎and the generalized order-3 Jacobsthal-Padovan orbit of the direct product $%‎ ‎D_{2n}\times {\mathbb{Z}}_{2m}$‎, ‎$\left( n,\;m\geq 3\right) $ and the‎ ‎semidirect product $D_{2n}\times _{\varphi }{\mathbb{Z}}_{2m}$‎, ‎$\left(‎ ‎n,\;m\geq 3\right) $.