Abstract
In this paper, we define the 2k-step Jordan-Fibonacci sequence, and then we study the 2k-step Jordan-Fibonacci sequence modulo m. Also, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the 2k-step Jordan-Fibonacci sequence when read modulo m, and we give the relationships among the orders of the cyclic groups obtained and the periods of the 2k-step Jordan-Fibonacci sequence modulo m. Furthermore, we extend the 2k-step Jordan-Fibonacci sequence to groups, and then we examine this sequence in the finite groups. Finally, we obtain the period of the 2k-step Jordan-Fibonacci sequence in the generalized quaternion group Q_{2^{n}} as applications of the results produced.
Highlights
Suppose that the (n + k)th term of a sequence is defined consecutively by a linear combination of the preceding k terms: an+k = c an + c an+ + · · · + ck– an+k, where c, c, . . . , ck– are real constants
In [ ], Kalman derived a number of closed-form formulas for the generalized sequence by the companion matrix method as follows
We study the k-step Jordan-Fibonacci sequence modulo m, and we obtain the cyclic groups from the multiplicative orders of the Jordan-Fibonacci matrix FJk of order k such that the elements of the matrix FJk when read modulo m
Summary
It is clear that the characteristic polynomial of the k-step Fibonacci sequence is as follows: Pk(x) = xk – xk– – · · · – x – . We study the k-step Jordan-Fibonacci sequence modulo m, and we obtain the cyclic groups from the multiplicative orders of the Jordan-Fibonacci matrix FJk of order k such that the elements of the matrix FJk when read modulo m. We derive the relationships among the orders of the cyclic groups obtained and the periods of the k-step Jordan-Fibonacci sequence modulo m.
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