In this paper, we have constituted 3-step general Fibonacci sequences in a nilpotent group with exponent p (p is a prime number) and nilpotency class 4 and given formulas to find the α term of the sequence. | Turk J Math 27 (2003) , 525 – 537. ¨ ITAK ˙ c TUB On General Fibonacci Sequences in Groups ¨ Engin Ozkan Abstract In this paper, we have constituted 3-step general Fibonacci sequences in a nilpotent group with exponent p (p is a prime number) and nilpotency class 4 and given formulas to find the α term of the sequence. Key Words: General Fibonacci sequences; nilpotent group; nilpotency class; fundamental period. 1. Introduction Let si denote the 3-step general recurrence defined by si = lsi−1 + msi−2 + nsi−3 for some l, m, n ∈ N. We assume that p does not divide n; then we get the definition of a 3-step general standard Fibonacci sequence as (0, 0, 1, l, l2 +m, l(l2 +m)+lm+n, .) in Z/pZ. If p were permitted to divide n, then the sequence would ultimately be periodic, but would never return to 0, 0, 1. This sequence or loop must be periodic and we use the letter k to denote the fundamental period of si that is the shortest period of that sequence. The fundamental period of a sequence satisfying a linear recurrence is sometimes called the Wall number of that sequence. Obviously k depends on p. In the recent years, there has been much interest in applications of Fibonacci numbers and sequences. Takahashi gives a fast algorithm which is based on the product of Lucas numbers to compute large Fibonacci numbers [8]. Fibonacci sequences have been an interesting subject in applied mathematics. West has shown by using transfer matrices AMS Subject Classification 2000: 11B39; 20D15 525 ¨ OZKAN that the number |Sn (123, 3214)| of permutations avoiding the patterns 123 and 3214 is the Fibonacci number F2n [11]. The study of Fibonacci sequences in groups began with the earlier work of Wall [10], where the ordinary Fibonacci sequences in cyclic groups were investigated. Vinson was particularly interested in ranks of apparition in ordinary Fibonacci sequences [9]. In the mid 1980’s, Wilcox extended the problem to abelian groups [12]. Prolific co-operation among .
