The Hadamard-type k-step Pell sequences in Finite Groups

In this work, we study the Hadamard-type k-step Pell sequence modulo m and then, we obtain the cyclic groups which are generated by the multiplicative orders of the Hadamard-type k-step Pell matrix when read modulo m. Then we extend the Hadamard-type k-step Pell sequence to groups and we redefine the Hadamard-type k-step Pell sequence by means of the elements of groups. Finally, we obtain the periods of the Hadamard-type ۳-step Pell sequence in the semi-dihedral group SD۲m and the quasi-dihedral group QD۲m.