Non-Abelian Sequenceable Groups Involving α-Covers

A non-abelian finite group is called sequenceable if for some positive integer , is -generated ( ) and there exist integers such that every element of is a term of the -step generalized Fibonacci sequence , , , . A remarkable application of this definition may be find on the study of random covers in the cryptography. The 2-step generalized sequences for the dihedral groups studied for their periodicity in 2006 by H. Aydin and it is proved that in many cases for and , they are not periodic. Aydin’s work was in continuation of the research works of R. Dikici (1997) and E. Ozkan (2003) where they studied the ordinary Fibonacci sequences (sequences without the powers) of elements of groups. In this paper we consider 3-step generalized Fibonacci sequences and prove that the quaternion group Q (for every integer ) and the dihedral group (for every integer ) are sequenceable. The -covers together with the Fibonacci lengths of the corresponding 3-step sequences have been calculated as well.