The Generalized Order-k Lucas Sequences in Finite Groups

Ömür Deveci1, 2 and Erdal Karaduman1,2 1 Department of Mathematics, Faculty of Arts and Science, Kafkas University, 36100 Kars, Turkey 2 Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey Correspondence should be addressed to Ömür Deveci, [email protected] Received 18 October 2011; Accepted 11 December 2011 Academic Editor: Reinaldo Martinez Palhares Copyright q 2012 Ö. Deveci and E. Karaduman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the generalized order-k Lucas sequences modulo m. Also, we define the ith generalized order-k Lucas orbit li,{α1 ,α2 ,...,αk−1} A G with respect to the generating set A and the constants α1, α2, and αk−1 for a finite groupG 〈A〉. Then, we obtain the lengths of the periods of the ith generalized order-k Lucas orbits of the binary polyhedral groups 〈n, 2, 2〉, 〈2, n, 2〉, 〈2, 2, n〉 and the polyhedral groups n, 2, 2 , 2, n, 2 , 2, 2, n for 1 ≤ i ≤ k.