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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 431840, 12 pages
http://dx.doi.org/10.1155/2011/431840
Research Article

The Polytopic--Step Fibonacci Sequences in Finite Groups

Department of Mathematics, Faculty of Art and Science, Kafkas University, 36100 Kars, Turkey

Received 22 May 2011; Revised 26 July 2011; Accepted 27 July 2011

Academic Editor: Binggen Zhang

Copyright © 2011 Ömür Deveci. 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.

Abstract

We study the polytopic--step Fibonacci sequences, the polytopic--step Fibonacci sequences modulo , and the polytopic--step Fibonacci sequences in finite groups. Also, we examine the periods of the polytopic--step Fibonacci sequences in semidihedral group .

1. Introduction

The well- known -step Fibonacci sequence is defined as Let    be a sequence of real numbers. A -generalized Fibonacci sequence is defined by the following linear recurrence relation of order : where are specified by the initial conditions.

The -step Fibonacci sequence, the -generalized Fibonacci sequence, and their properties have been studied by several authors; see, for example, [15].

The -step Fibonacci sequence is a special case of a sequence which is defined as a linear combination by Kalman as follows where are real constants. In [6], Kalman derived a number of closed-form formulas for the generalized sequence by companion matrix method as follows: Then, by an inductive argument he obtained A sequence of group elements is periodic if, after a certain point, it consists only of repetitions of a fixed subsequence. The number of elements in the repeating subsequence is called the period of the sequence. For example, the sequence is periodic after the initial element and has period 4. A sequence of group elements is simply periodic with period if the first elements in the sequence form a repeating subsequence. For example, the sequence is simply periodic with period 6.

Definition 1.1. For a finitely generated group , where , the sequence , , , , is called the Fibonacci orbit of with respect to the generating set , denoted by . If is periodic, then the length of the period of the sequence is called the Fibonacci length of with respect to generating set , written as [7].

Definition 1.2. For every integer , where , the sequence of the elements of defined by is called a -step generalized Fibonacci sequence of , for some positive integers [8].

Definition 1.3. A -nacci sequence in a finite group is a sequence of group elements for which, given an initial (seed) set , each element is defined by We also require that the initial elements of the sequence, , generate the group, thus forcing the -nacci sequence to reflect the structure of the group. The -nacci sequence of a group seeded by is denoted by and its period is denoted by [9].

The Fibonacci sequence, the -nacci sequence, and the generalized order- Pell sequence in finite groups have been studied by some authors, and different periods of these sequences in different finite groups have been obtained; see, for example, [7, 916]. Formulas which classified according to certain rules for this periods are critical to be used in cryptography, see, for example, [1719]. Because the exponents of each term in the generalized Fibonacci sequence are determined randomly, classification according to certain rule of periods is resulting from application of this sequence in groups is possible, only if the exponent of each term are determined integers obtained according to a certain rule. Therefore, In this paper, by expanding the -step Fibonacci sequence which is special type of the generalized Fibonacci sequences with polytopic numbers which are a well-known family of integers, we conveyed the sequence named the polytopic--step Fibonacci sequence that exponent of term is determined that formula to finite groups and named the polytopic--step Fibonacci sequence in finite groups as polytopic--nacci sequence. Because of varying both and according to the number of step and the exponent of each term of this is determined according to a certain rule, the polytopic--step Fibonacci sequence is more useful and more general than the -nacci sequences and the generalized order- Pell sequence which varying only by the number of step. So that considered by different value, different step values and different initial (seed) sets, different lineer recurrence sequences which are a special type of generalized Fibonacci sequences occur, and thus by conveying the polytopic--step Fibonacci sequence to finite groups, more useful and more general formulas than formulas used to obtain periods of the -nacci and the generalized order- Pell sequence in finite groups are obtained to be used in cryptography.

In this paper, the usual notation is used for a prime number.

2. The Polytopic--Step Fibonacci Sequences

The well-known -topic numbers are defined as When , the -topic numbers, , are reduced to the triangular numbers. In [20], Gandhi and Reddy obtained triangular numbers in the generalized Pell sequence and generalized associated Pell sequence which are defined for a fixed , respectively, as Now we define for a fixed integer , a new sequence called the polytopic--step Fibonacci sequence , by Obviously, if we take in (2.3), then this sequence reduces to the well-known -step Fibonacci sequence. When and in (2.3), we call the polytopic Fibonacci sequence.

By (2.3), we can write for the polytopic--step Fibonacci sequence. Let The matrix is called the polytopic--step Fibonacci matrix.

We obtain that the polytopic Fibonacci sequences are generated by a matrix for a fixed integer : which can be proved by mathematical induction.

3. The Polytopic--Step Fibonacci Sequences Modulo

In this section we examine the polytopic--step Fibonacci sequences modulo for and .

Reducing the polytopic--step Fibonacci sequence by a modulus , we can get a repeating sequence denoted by where . It has the same recurrence relation as in (2.3).

Theorem 3.1. is a periodic sequence for and .

Proof. Let . Then we have that is finite, that is, for any , there exist such that . From the definition of the polytopic--step Fibonacci sequence we have , that is, Then we can easily get that ,   and , which implies that is a periodic sequence.
Let denote the smallest period of , called the period of the polytopic--step Fibonacci sequence modulo . When , is the period of the polytopic Fibonacci sequence modulo .

Example 3.2. We have and then repeat. So we get .

By elementary number theory it is easy to prove that if , , where ’s are distinct primes, then .

For a given matrix with ’s being integers, means that every entry of is reduced modulo , that is, . Let be a cyclic group, and let denote the order of with (where by we mean that is not divided by ) and the transpose of a matrix. It is clear that We then obtain that is least positive integer such that

Theorem 3.3. Let . If , then .

Proof. It is clear that is divisible by . Then we need only to prove that is divisible by . Let . Then we have The elements of the matrix are in the following forms: We thus obtain that So we get that , which yields that is divisible by . We are done.

Theorem 3.4. Let , and let be the largest positive integer such that . Then for every . In particular, if , then holds for every .

Proof. Let be a positive integer. Since , that is, , we get that is divided by . On the other hand, writing , we have which yields that is divided by . Therefore, or , and the latter holds if, and only if, there is an which is not divisible by . Since , there is an which is not divisible by , thus, . The proof is finished by induction on .

Conjecture 3.5. Let . If , then there exists a with such that is divided by .

Table 1 list some primes for which the conjecture is true when and .

tab1
Table 1: The length of .

4. The Polytopic--Nacci Sequences in Finite Groups

Definition 4.1. For a finitely generated group , where , we define the polytopic Fibonacci orbit with respect to the generating set to be the sequence of the elements of such that

Example 4.2. Let , where . is

Definition 4.3. A polytopic--nacci sequence in a finite group is a sequence of group elements for which, given an initial (seed) set , each element is defined by It is required that the initial elements of the sequence, , generate the group, thus, forcing the polytopic--nacci sequence to reflect the structure of the group. We denote the polytopic--nacci sequence of a group generated by by .

Example 4.4. Let , where . is It is important to note that the polytopic Fibonacci orbit of a -generated group is a polytopic--nacci sequence.
The classic polytopic Fibonacci sequence in the integers modulo can be written as . We call a polytopic-2-nacci sequence of a group of elements a polytopic Fibonacci sequence of a finite group.

Theorem 4.5. A polytopic--nacci sequence in a finite group is periodic.

Proof. The proof is similar to the proof of Theorem  1 in [6] and is omitted.
We denote the period of a polytopic--nacci sequence by . When , and are reduced to and , respectively.
From the definition, it is clear that the period of a polytopic--nacci sequence in a finite group depends on the chosen generating set and the order in which the assignments of are made.

Definition 4.6. Let be a finite group. If there exists a polytopic--nacci sequence of the group such that every element of the group appears in the sequence, then the group is called polytopic--nacci sequenceable.

It is important to note that the direct product of polytopic--nacci sequenceable groups is not necessarily polytopic--nacci sequenceable. Consider that the group is defined by the presentation The polytopic Fibonacci sequences of the group for are Since the elements do not in either sequences, the group is not polytopic-2-nacci sequenceable.

The group has a polytopic Fibonacci sequence and hence is polytopic-2-nacci sequenceable. The group has a polytopic Fibonacci sequence and hence is polytopic-2-nacci sequenceable.

We will now address the periods of the polytopic--nacci sequences in specific classes of groups. A group is semidihedral group of order if for every . Note that the orders and are and 2, respectively.

Theorem 4.7. The periods of the polytopic--nacci sequences in the group for initial (seed) set, , and are as follows:(i). (ii).

Proof. (i) If , we have the polytopic-2-nacci sequence for By mathematical induction, it is easy to prove that So we get , It is easy to see that . Since the elements succeeding , , depend on and for their values, the cycle begins again with the , that is, and . Thus, the period of is .
If , we have the polytopic-3-nacci sequence for : By mathematical induction, it is easy to prove that , So we get , , . It is easy to see that . Since the elements succeeding depend on , , and for their values, the cycle begins again with , that is , and . Thus, the period of is . The proof for is similar and is omitted.
(ii) If , we have the polytopic--nacci sequence for where are natural numbers and are even natural numbers. So we need the smallest such that . If we choose , we obtain , , , , , , , , since . So we get for .

Theorem 4.8. The periods of the the polytopic--nacci sequences in the group for initial (seed) sets , and are as follows:(i), (ii).

Proof. The proof is similar to the proof of Theorem 4.5 and is omitted.

Acknowledgment

This project was supported by the Commission for the Scientific Research Projects of Kafkas University. The Project no. 2010-FEF-61.

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