On the Basic k-nacci Sequences in Finite Groups

Abstract

We define the basic k-nacci sequences and the basic periods of these sequences in finite groups, then we obtain the basic periods of the basic k-nacci sequences and the periods of the k-nacci sequences in symmetric group 𝑆₄, its subgroups, and binary polyhedral groups which related with these groups.

1. Introduction

The study of Fibonacci sequences in groups began with the earlier work of Wall [1], where the ordinary Fibonacci sequences in cyclic groups were investigated. In the mid-eighties, Wilcox extended the problem to Abelian groups [2]. The theory is expanded to some finite simple groups by Campbell et al. [3]. There, they defined the Fibonacci length of the Fibonacci orbit and the basic Fibonacci length of the basic Fibonacci orbit in a 2-generator group. The concept of Fibonacci length for more than two generators has also been considered; see, for example, [4, 5]. Also, the theory has been expanded to the nilpotent groups; see, for example, [6, 7]. Other works on Fibonacci length are discussed in, for example, [8–10]. Knox proved that the periods of k-nacci (k-step Fibonacci) sequences in dihedral groups were equal to 2𝑘+2 [11]. Deveci, Karaduman, and Campbell examined the period of the k-nacci sequences in some finite binary polyhedral groups in [12]. Recently, k-nacci sequences have been investigated; see, for example, [13, 14].

This paper defines the basic k-nacci sequences and the periods of these sequences in finite groups and discusses the basic periods of the basic k-nacci sequences and the periods of the k-nacci sequences in the symmetric group 𝑆₄, alternating group 𝐴₄, 𝐷₂ four-group, and binary polyhedral groups ⟨2,3,4⟩ and ⟨2,3,3⟩ with related 𝑆₄ and 𝐴₄, respectively. We consider the groups 𝑆₄, 𝐴₄, ⟨2,3,4⟩, and ⟨2,3,3⟩ both as 2-generator and as 3-generator groups.

A k-nacci sequence in a finite group is a sequence of group elements 𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑛,… for which, given an initial (seed) set 𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑗−1, each element is defined by𝑥_𝑛=𝑥₀𝑥₁⋯𝑥_𝑛−1𝑥for𝑗≤𝑛<𝑘,_𝑛−𝑘𝑥_{𝑛−𝑘+1}⋯𝑥_𝑛−1for𝑛≥𝑘. (1.1) We also require that the initial elements of the sequence 𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑗−1 generate the group, thus forcing the k-nacci sequence to reflect the structure of the group. The k-nacci sequence of a group 𝐺 generated by 𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑗−1 is denoted by 𝐹_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1) [11].

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 𝑎,𝑏,𝑐,𝑑,𝑒,𝑏,𝑐,𝑑,𝑒,𝑏,𝑐,𝑑,𝑒,… is periodic after the initial element 𝑎 and has period 4. A sequence of group elements is simply periodic with period k if the first k elements in the sequence form a repeating subsequence. For example, the sequence 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,… is simply periodic with period 6. In [11], Knox had denoted the period of a k-nacci sequence 𝐹_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1) by 𝑃_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1).

Definition 1.1. For a finitely generated group 𝐺=⟨𝐴⟩, where 𝐴={𝑎₁,𝑎₂,…,𝑎_𝑛}, the sequence 𝑥_𝑖=𝑎_𝑖+1, 0≤𝑖≤𝑛−1, 𝑥_𝑖+𝑛=∏^𝑛_𝑗=1𝑥_{𝑖+𝑗−1}, 𝑖≥0 is called the Fibonacci orbit of 𝐺 with respect to the generating set 𝐴, denoted as 𝐹_𝐴(𝐺) [4].

Definition 1.2. If 𝐹_𝐴(𝐺) is simply periodic, then the period of the sequence is called the Fibonacci length of 𝐺 with respect to generating set 𝐴, written, 𝐿𝐸𝑁_𝐴(𝐺) [4].
Notice that the orbit of a k-generated group is a k-nacci sequence.
Let 𝐺 be a finite 𝑗-generator group, and let 𝑋 be the subset of 𝐺×𝐺×𝐺⋯×𝐺_𝑗 such that (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋 if and only if 𝐺 is generated by 𝑥₀,𝑥₁,…,𝑥_𝑗−1. We call (𝑥₀,𝑥₁,…,𝑥_𝑗−1) a generating 𝑗-tuple for 𝐺.

2. Basic Period of Basic k-nacci Sequence

To examine the concept more fully, we study the action of automorphism group Aut𝐺 of 𝐺 on 𝑋 and on the k-nacci sequences 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1), (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋. Now, Aut𝐺 consists of all isomorphism 𝜃∶𝐺→𝐺 and if 𝜃∈ Aut𝐺 and (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋, then (𝑥₀𝜃,𝑥₁𝜃,…,𝑥_𝑗−1𝜃)∈𝑋.

For a subset 𝐴⊆𝐺 and 𝜃∈ Aut𝐺, the image of 𝐴 under 𝜃 is 𝐴𝜃={𝑎𝜃∶𝑎∈𝐴}. (2.1)

Definition 2.1. For a generating pair (𝑥,𝑦)∈𝑋, the basic Fibonacci orbit 𝐹_𝑥,𝑦 of the basic length 𝑚 is defined by the sequence {𝑏_𝑖} of elements of 𝐺 such that𝑏₀=𝑥,𝑏₁=𝑦,𝑏_𝑖+2=𝑏_𝑖𝑏_𝑖+1,𝑖≥0, (2.2) where 𝑚≥1 is the least integer with 𝑏₀=𝑏_𝑚𝜃,𝑏₁=𝑏_𝑚+1𝜃, (2.3) for some 𝜃∈ Aut𝐺. Since 𝑏_𝑚,𝑏_𝑚+1 generate 𝐺, it follows that 𝜃 is uniquely determined. For more information, see [3].

Lemma 2.2. Let (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋 and let 𝜃∈ Aut 𝐺, then (𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1))𝜃=𝐹_𝑘(𝐺∶𝑥₀𝜃,𝑥₁𝜃,…𝑥_𝑗−1𝜃).

Proof. Let 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1)={𝑏_𝑖}. The result is obvious since {𝑏_𝑖}𝜃={𝑏_𝑖𝜃} and𝑏_𝑖+𝑘𝜃=𝑏_𝑖𝑏_𝑖+1⋯𝑏_{𝑖+𝑘−1}𝜃=𝑏_𝑖𝜃𝑏_𝑖+1𝜃⋯𝑏_{𝑖+𝑘−1}𝜃. (2.4) Each generating 𝑗-tuple (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋 maps to |Aut𝐺| distinct elements of 𝑋 under the action of elements of Aut𝐺. Hence, there are 𝑑_𝑗(𝐺)=|𝑋|/|Aut𝐺|, (2.5) (where |𝑋| means the number of elements of 𝑋) nonisomorphic generating 𝑗-tuples for 𝐺. The notation 𝑑_𝑗(𝐺) was introduced in [15].
Suppose that 𝜔 elements of Aut𝐺 map 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1) into itself, then there are |Aut𝐺|/𝜔 distinct k-nacci sequences 𝐹_𝑘(𝐺∶𝑥₀𝜃,𝑥₁𝜃,…𝑥_𝑗−1𝜃) for 𝜃∈ Aut𝐺.

Definition 2.3. For a 𝑗-tuple (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋, the basic k-nacci sequence 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1) of the basic period 𝑚 is a sequence of group elements 𝑏₀,𝑏₁,𝑏₂,…,𝑏_𝑛,… for which, given an initial (seed) set 𝑏₀=𝑥₀,𝑏₁=𝑥₁,𝑏₂=𝑥₂,…,𝑏_𝑗−1=𝑥_𝑗−1, each element is defined by𝑏_𝑛=𝑏₀𝑏₁⋯𝑏_𝑛−1𝑏for𝑗≤𝑛<𝑘,_𝑛−𝑘𝑏_{𝑛−𝑘+1}⋯𝑏_𝑛−1for𝑛≥𝑘, (2.6) where 𝑚≥1 is the least integer with 𝑏₀=𝑏_𝑚𝜃,𝑏₁=𝑏_𝑚+1𝜃,𝑏₂=𝑏_𝑚+2𝜃,…,𝑏_𝑘−1=𝑏_{𝑚+𝑘−1}𝜃, (2.7) for some 𝜃∈ Aut𝐺. Since 𝐺 is a finite 𝑗-generator group and 𝑏_𝑚, 𝑏_𝑚+1, …, 𝑏_{𝑚+𝑗−1} generate 𝐺, it follows that 𝜃 is uniquely determined. The basic k-nacci sequence 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1) is finite containing 𝑚 element.
In this paper, we denote the basic period of the basic k-nacci sequence 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1) by 𝐵𝑃_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1).
From the definitions, it is clear that the periods of the k-nacci sequences and the basic k-nacci sequences in a finite group depend on the chosen generating set and the order of the generating elements.

Theorem 2.4. Let 𝐺 be a finite group and (𝑥₀,𝑥₁,…,𝑥_𝑗−1)∈𝑋. If 𝑃_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1)=𝑛 and 𝐵𝑃_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1)=𝑚, then 𝑚 divides 𝑛, and there are 𝑛/𝑚 elements of Aut𝐺 which map 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1) into itself.

Proof. We have 𝑛=𝑚𝜆 where 𝜆 is the order of automorphism 𝜃∈ Aut𝐺 since𝐹_𝑘𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1=𝐹_𝑘𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1∪𝐹_𝑘𝐺∶𝑥₀𝜃,𝑥₁𝜃,…𝑥_𝑗−1𝜃∪𝐹_𝑘𝐺∶𝑥₀𝜃²,𝑥₁𝜃²,…𝑥_𝑗−1𝜃²∪⋯ (2.8) and 𝐵𝑃_𝑘(𝐺;𝑥₀,𝑥₁,…,𝑥_𝑗−1)=𝐵𝑃_𝑘(𝐺;𝑥₀𝜃,𝑥₁𝜃,…,𝑥_𝑗−1𝜃). Clearly, 1,𝜃,𝜃²,…,𝜃^𝜆−1 map 𝐹_𝑘(𝐺∶𝑥₀,𝑥₁,…𝑥_𝑗−1) into itself.

3. Applications

Definition 3.1. The polyhedral group (𝑙,𝑚,𝑛) for 𝑙,𝑚,𝑛>1 is defined by the presentation𝑥,𝑦,𝑧∶𝑥^𝑙=𝑦^𝑚=𝑧^𝑛=𝑥𝑦𝑧=𝑒, (3.1) or 𝑥,𝑦∶𝑥^𝑙=𝑦^𝑚=𝑥𝑦^𝑛=𝑒. (3.2) The polyhedral group (𝑙,𝑚,𝑛) is finite if and only if the number 1𝜇=𝑙𝑚𝑛𝑙+1𝑚+1𝑛−1=𝑚𝑛+𝑛𝑙+𝑙𝑚−𝑙𝑚𝑛 (3.3) is positive, that is, in the cases (2,2,𝑛), (2,3,3), (2,3,4), and (2,3,5). Its order is 2𝑙𝑚𝑛/𝜇. 𝐴₄, 𝑆₄, and 𝐴₅ are the groups (2,3,3), (2,3,4), and (2,3,5), respectively. Also, the groups 𝐴₄, 𝑆₄, and 𝐴₅ being isomorphic to the groups of rotations of the regular tetrahedron, octahedron, and icosahedron. Using Tietze transformations, we may show that (𝑙,𝑚,𝑛)≅(𝑚,𝑛,𝑙)≅(𝑛,𝑙,𝑚). For more information on these groups, see [16] and [17, pp. 67-68].

Definition 3.2. The binary polyhedral group ⟨𝑙,𝑚,𝑛⟩, for 𝑙,𝑚,𝑛>1, is defined by the presentation𝑥,𝑦,𝑧∶𝑥^𝑙=𝑦^𝑚=𝑧^𝑛=𝑥𝑦𝑧, (3.4) or 𝑥,𝑦∶𝑥^𝑙=𝑦^𝑚=𝑥𝑦^𝑛. (3.5) The binary polyhedral group ⟨𝑙,𝑚,𝑛⟩ is finite if and only if the number 𝑘=𝑙𝑚𝑛(1/𝑙+1/𝑚+1/𝑛−1)=𝑚𝑛+𝑛𝑙+𝑙𝑚−𝑙𝑚𝑛 is positive. Its order is 4𝑙𝑚𝑛/𝑘.
For more information on these groups, see [17, pp. 68–71].

Definition 3.3. Let 𝑓_𝑛^(𝑘) denote the 𝑛th member of the 𝑘-step Fibonacci sequence defined as𝑓_𝑛^(𝑘)=^𝑘_𝑗=1𝑓^(𝑘)_𝑛−𝑗for𝑛>𝑘, (3.6) with boundary conditions 𝑓_𝑖^(𝑘)=0 for 1≤𝑖<𝑘 and 𝑓_𝑘^(𝑘)=1. Reducing this sequence by a modulo 𝑚, we can get a repeating sequence, which we denote by 𝑓(𝑘,𝑚)=𝑓₁^(𝑘,𝑚),𝑓₂^((𝑘,𝑚),…,𝑓_𝑛^(𝑘,𝑚)…, (3.7) where 𝑓_𝑖^(𝑘,𝑚)=𝑓_𝑖^(𝑘)(mod𝑚). We then have that (𝑓₁^(𝑘,𝑚),𝑓₂^(𝑘,𝑚),…,𝑓_𝑘^(𝑘,𝑚))=(0,0,…0,1), and it has the same recurrence relation as in (3.6) [18].

Theorem 3.4 (𝑓(𝑘,𝑚) is a periodic sequence [18]). Let ℎ_𝑘(𝑚) denote the smallest period of 𝑓(𝑘,𝑚), called the period of 𝑓(𝑘,𝑚) or the wall number of the k-step Fibonacci sequence modulo 𝑚.

Theorem 3.5. The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the group 𝑆₄ are as follows.
if the group is defined by the presentation 𝑆₄=⟨𝑥,𝑦,𝑧∶𝑥²=𝑦³=𝑧⁴=𝑥𝑦𝑧=𝑒⟩, then
(i)if 𝑘=2,𝑃₂(𝑆₄;𝑦,𝑧,𝑥)=18 and 𝐵𝑃₂(𝑆₄;𝑦,𝑧,𝑥)=9,(ii)if 𝑘>2,𝑃_𝑘(𝑆₄;𝑥,𝑦,𝑧)=6𝑘+6 and 𝐵𝑃_𝑘(𝑆₄;𝑥,𝑦,𝑧)=3𝑘+3. If 𝑆₄ has the presentation 𝑆₄=⟨𝑥,𝑦∶𝑥²=𝑦³=(𝑥𝑦)⁴=𝑒⟩, then(i′) if 𝑘=2,𝑃₂(𝑆₄;𝑥,𝑦)=18 and 𝐵𝑃₂(𝑆₄;𝑥,𝑦)=9,(ii′) if 𝑘>2, 𝑃_𝑘(𝑆₄;𝑥,𝑦)=6𝑘+6 and 𝐵𝑃_𝑘(𝑆₄;𝑥,𝑦)=3𝑘+3.

Proof. Firstly, let us consider the 3-generator case. We first note that |𝑥|=2, |𝑦|=3, and |𝑧|=4 (where |𝑥| means the order of 𝑥).
(i)If 𝑘=2, we have the sequence for the generating triple (𝑦, 𝑧, 𝑥),𝑦,𝑧,𝑥,𝑦²,𝑥𝑦²,𝑦²𝑥𝑦²,𝑧²𝑦,𝑧²𝑦𝑧³𝑦,𝑦𝑥𝑦,𝑥𝑦𝑥,𝑥𝑦²,𝑥,𝑥𝑦²𝑥,𝑦²𝑥,𝑦𝑥𝑦,𝑦𝑥𝑧,𝑧𝑦,𝑦²𝑥𝑦²,𝑦,𝑧,𝑥,…, (3.8) which has period 18 and the basic period 9 since 𝑥𝜃=𝑥, 𝑦𝜃=𝑥𝑦𝑥, and 𝑧𝜃=𝑥𝑦², where 𝜃 is the inner automorphism induced by conjugation by 𝑥. (ii)If 𝑘=3, we have the sequence for the generating triple (𝑥, 𝑦, 𝑧),𝑥,𝑦,𝑧,𝑒,𝑥,𝑦²,𝑥𝑦²,𝑥𝑧𝑦²,𝑥,𝑦,𝑦𝑥𝑦²,𝑥𝑧𝑦²𝑦,𝑥,₂,𝑦𝑥,𝑒,𝑥,𝑦,𝑥𝑦,𝑧²,𝑥,𝑦²𝑧𝑦,𝑧²,𝑥,𝑦,𝑧…, (3.9) which has period 24 and the basic period 12 since 𝑥𝜃=𝑥, 𝑦𝜃=𝑦², and 𝑧𝜃=𝑦𝑥 where 𝜃 is an outer automorphism of order 2.
If 𝑘≥4, the first 𝑘 elements of sequence for the generating triple (𝑥, 𝑦, 𝑧) are𝑥₀=𝑥,𝑥₁=𝑦,𝑥₂=𝑧,𝑥₃=𝑥𝑦𝑧,𝑥₄=𝑥𝑦𝑧²…,𝑥_𝑘−1=𝑥𝑦𝑧^2𝑘−4. (3.10) Thus, using the above information, sequence reduces to 𝑥₀=𝑥,𝑥₁=𝑦,𝑥₂=𝑧,𝑥₃=𝑒,…,𝑒,𝑥_𝑘−1=𝑒, (3.11) where 𝑥_𝑗=𝑒 for 3≤𝑗≤𝑘−1. Thus, 𝑥_𝑘=𝑒,𝑥_𝑘+1=𝑥,𝑥_𝑘+2=𝑦²,𝑥_𝑘+3=𝑥𝑦²,𝑥_𝑘+4=𝑥𝑧𝑦²,𝑥_𝑘+5=𝑒,…,𝑒,𝑥_2𝑘+1=𝑒,𝑥_2𝑘+2=𝑥,𝑥_2𝑘+3𝑥=𝑦,_2𝑘+4=𝑦𝑥𝑦²,𝑥_2𝑘+5=𝑥𝑧𝑦²,𝑥_2𝑘+6=𝑒,…,𝑒,𝑥_3𝑘+2𝑥=𝑒,_3𝑘+3=𝑥,𝑥_3𝑘+4=𝑦²,𝑥_3𝑘+5=𝑦𝑥,𝑥_3𝑘+6=𝑒,…,𝑒,𝑥_4𝑘+3𝑥=𝑒,_4𝑘+4=𝑥,𝑥_4𝑘+5=𝑦,𝑥_4𝑘+6=𝑥𝑦,𝑥_4𝑘+7=𝑧²,𝑥_4𝑘+8=𝑒,…,𝑒,𝑥_5𝑘+4=𝑒,𝑥_5𝑘+5=𝑥,𝑥_5𝑘+6=𝑦²,𝑥_5𝑘+7=𝑧𝑦,𝑥_5𝑘+8=𝑧²,𝑥_5𝑘+9=𝑒,…,𝑒,𝑥_6𝑘+5=𝑒, (3.12) where 𝑥_𝑗=𝑒 for 𝑘+5≤𝑗≤2𝑘+1, 2𝑘+6≤𝑗≤3𝑘+2, 3𝑘+6≤𝑗≤4𝑘+3, 4𝑘+8≤𝑗≤5𝑘+4, and 5𝑘+9≤𝑗≤6𝑘+5.
We also have𝑥_6𝑘+6=^6𝑘+5_𝑖=5𝑘+6𝑥_𝑖=𝑥,𝑥_6𝑘+7=^6𝑘+6_𝑖=5𝑘+7𝑥_𝑖=𝑦,𝑥_6𝑘+8=^6𝑘+7_𝑖=5𝑘+8𝑥_𝑖=𝑧. (3.13) Since the elements succeeding 𝑥_6𝑘+6, 𝑥_6𝑘+7, and 𝑥_6𝑘+8 depend on 𝑥, 𝑦, and 𝑧 for their values, the cycle begins again with the 6𝑘+6^th element, that is, 𝑥₀=𝑥_6𝑘+6,𝑥₁=𝑥_6𝑘+7,𝑥₂=𝑥_6𝑘+8,…. Thus, 𝑃_𝑘(𝑆₄;𝑥,𝑦,𝑧)=6𝑘+6.
It is easy to see from the above sequence that𝑥_3𝑘+3=𝑥,𝑥_3𝑘+4=𝑦²,𝑥_3𝑘+5=𝑦𝑥,𝑥_3𝑘+6=𝑒,…,𝑒,𝑥_4𝑘+2=𝑒. (3.14)𝐵𝑃_𝑘(𝑆₄;𝑥,𝑦,𝑧)=3𝑘+3 since 𝑥𝜃=𝑥, 𝑦𝜃=𝑦², and 𝑧𝜃=𝑦𝑥 where 𝜃 is an outer automorphism of order 2.
Secondly, let us consider the 2-generator case. We first note that |𝑥|=2, |𝑦|=3, and |𝑥𝑦|=4.
(i′) If 𝑘=2,𝑃₂(𝑆₄;𝑥,𝑦)=18 and 𝐵𝑃₂(𝑆₄;𝑥,𝑦)=9 since 𝑥𝜃=𝑥 and 𝑦𝜃=𝑥𝑦𝑥 where 𝜃 is the inner automorphism induced by conjugation by 𝑥.(ii′)If 𝑘>2, 𝑃_𝑘(𝑆₄;𝑥,𝑦)=6𝑘+6 and 𝐵𝑃_𝑘(𝑆₄;𝑥,𝑦)=3𝑘+3 since 𝑥𝜃=𝑥 and 𝑦𝜃=𝑦² where 𝜃 is an outer automorphism of order 2. The proofs are similar to above and are omitted.

Theorem 3.6. The periods of the k-nacci sequences and the basic periods of the basic k-nacci sequences in the binary polyhedral group ⟨2,3,4⟩ are as follows.
If the group is defined by the presentation ⟨2,3,4⟩=⟨𝑥,𝑦,𝑧∶𝑥²=𝑦³=𝑧⁴=𝑥𝑦𝑧⟩, then
(i)if 𝑘=2,𝑃₂(⟨2,3,4⟩;𝑦,𝑧,𝑥)=18 and 𝐵𝑃₂(⟨2,3,4⟩;𝑦,𝑧,𝑥)=9,(ii)if 𝑘>2,𝑃_𝑘(⟨2,3,4⟩;𝑥,𝑦,𝑧)=6𝑘+6 and 𝐵𝑃_𝑘(⟨2,3,4⟩;𝑥,𝑦,𝑧)=6𝑘+6. If the group is defined by the presentation ⟨2,3,4⟩=⟨𝑥,𝑦∶𝑥²=𝑦³=(𝑥𝑦)⁴⟩, then(i′) if 𝑘=2,𝑃₂(⟨2,3,4⟩;𝑥,𝑦)=18 and 𝐵𝑃₂(⟨2,3,4⟩;𝑥,𝑦)=9,(ii′) if 𝑘>2, 𝑃_𝑘(⟨2,3,4⟩;𝑥,𝑦)=6𝑘+6 and 𝐵𝑃_𝑘(⟨2,3,4⟩;𝑥,𝑦)=6𝑘+6.

Proof. Firstly, let us consider the 2-generator case. We first note that |𝑥|=4, |𝑦|=6, and |𝑥𝑦|=8.
(i′) If 𝑘=2, we have the sequence for the generating pair (𝑥, 𝑦),𝑥,𝑦,𝑥𝑦,𝑦𝑥𝑦,𝑥𝑦²𝑥𝑦,𝑥𝑦𝑥𝑦²𝑥,𝑦²𝑥𝑦²,𝑥𝑦⁵𝑥,𝑥𝑦,𝑥³,𝑥𝑦𝑥³,𝑦𝑥³,𝑦²𝑥𝑦²,𝑦²𝑥𝑦𝑥,𝑦𝑥𝑦²,𝑦𝑥𝑦,𝑦²,𝑦⁴𝑥,𝑥,𝑦,…, (3.15) which has period 18 and the basic period 9 since 𝑥𝜃=𝑥³ and 𝑦𝜃=𝑥³𝑦𝑥 where 𝜃 is a outer automorphism of order 2. (ii′) If 𝑘=3, we have the sequence for the generating pair (𝑥, 𝑦),𝑥,𝑦,𝑥𝑦,𝑥𝑦²,𝑥,𝑦²,𝑦⁵𝑥𝑦,𝑥𝑦²,𝑥,𝑦,𝑥𝑦³,𝑥𝑦⁴,𝑥³,𝑦²,𝑥𝑦²,𝑦𝑥²,𝑥³,𝑦,𝑦𝑥𝑦²,𝑦𝑥²,𝑥³,𝑦²,𝑦⁴𝑥,𝑒,𝑥,𝑦,𝑥𝑦,…, (3.16) which has period 24 and the basic period 24 since 𝑥𝜃=𝑥 and 𝑦𝜃=𝑦 where 𝜃 is an inner automorphism induced by conjugation by 𝑥².
If 𝑘=4, we have the sequence for the generating pair (𝑥, 𝑦),𝑥,𝑦,𝑥𝑦,𝑥𝑦²,𝑥𝑦⁴,𝑥³,𝑦²,𝑦⁵𝑥𝑦,𝑥𝑦²,𝑒,𝑥,𝑦,𝑥𝑦³,𝑥𝑦⁴,𝑒,𝑥³,𝑦²,𝑥𝑦²,𝑦𝑥²,𝑥²,𝑥,𝑦,𝑦𝑥𝑦²,𝑦𝑥²,𝑒,𝑥³,𝑦²,𝑦⁴𝑥,𝑒,𝑒,𝑥,𝑦,𝑥𝑦,𝑥𝑦²,…, (3.17) which has period 30 and the basic period 30 since 𝑥𝜃=𝑥 and 𝑦𝜃=𝑦 where 𝜃 is an inner automorphism induced by conjugation by 𝑥².
If 𝑘≥5, the first 𝑘 elements of sequence for the generating pair (𝑥, 𝑦) are𝑥₀=𝑥,𝑥₁=𝑦,𝑥₂=𝑥𝑦,𝑥₃=𝑥𝑦²,𝑥₄=𝑥𝑦⁴,𝑥₅=𝑥𝑦⁸…,𝑥_𝑘−1=𝑥𝑦^2𝑘−3. (3.18) Thus, using the above information, sequence reduces to 𝑥₀=𝑥,𝑥₁=𝑦,𝑥₂=𝑥𝑦,𝑥₃=𝑥𝑦²,𝑥₄=𝑥𝑦⁴,𝑥₅=𝑒,…,𝑒,𝑥_𝑘−1=𝑒, (3.19) where 𝑥_𝑗=𝑒for5≤𝑗≤𝑘−1. Thus, 𝑥_𝑘=𝑒,𝑥_𝑘+1=𝑥³,𝑥_𝑘+2=𝑦²,𝑥_𝑘+3=𝑦⁵𝑥𝑥𝑦,_𝑘+4=𝑥𝑦²,𝑥_𝑘+5=𝑒,…,𝑒,𝑥_2𝑘+1=𝑒,𝑥_2𝑘+2𝑥=𝑥,_2𝑘+3=𝑦,𝑥_2𝑘+4=𝑥𝑦³,𝑥_2𝑘+5=𝑥𝑦⁴,𝑥_2𝑘+6𝑥=𝑒,…,𝑒,_3𝑘+2=𝑒,𝑥_3𝑘+3=𝑥³,𝑥_3𝑘+4=𝑦²,𝑥_3𝑘+5=𝑥𝑦²,𝑥_3𝑘+6=𝑦𝑥²,𝑥_3𝑘+7=𝑥²,𝑥_3𝑘+8=𝑒…,𝑒,𝑥_4𝑘+3𝑥=𝑒,_4𝑘+4=𝑥,𝑥_4𝑘+5=𝑦,𝑥_4𝑘+6=𝑦𝑥𝑦²𝑥_4𝑘+7=𝑦𝑥²,𝑥_4𝑘+8=𝑒,…,𝑒,𝑥_5𝑘+4=𝑒,𝑥_5𝑘+5=𝑥³,𝑥_5𝑘+6=𝑦²,𝑥_5𝑘+7=𝑦⁴𝑥,𝑥_5𝑘+8=𝑒,…,𝑒,𝑥_6𝑘+5=𝑒, (3.20)(3.21) where 𝑣₁,𝑣₂∈𝑁, and ℎ_𝑘(6) denote the wall number of the 𝑘-step Fibonacci sequence modulo 6.
If the group is defined by the presentation ⟨2,3,3⟩=⟨𝑥,𝑦∶𝑥²=𝑦³=(𝑥𝑦)³⟩, then
(i^′) if 𝑘=2, 𝑃₂(⟨2,3,3⟩;𝑥,𝑦)=48 and 𝐵𝑃₂(⟨2,3,3⟩;𝑥,𝑦)=12,(ii^′) if 𝑘>2, 𝑃_𝑘⟨2,3,3⟩;𝑥,𝑦=3𝐵𝑃_𝑘⟨2,3,3⟩;𝑥,𝑦,𝑘≡0mod4,𝐵𝑃_𝑘⟨2,3,3⟩;𝑥,𝑦,𝑘≢0mod4, (3.33)𝐵𝑃_𝑘⟨⟩=𝑣2,3,3;𝑥,𝑦¹ℎ_𝑘(6)𝑣,𝑘≡0mod4,₂ℎ_𝑘(6),𝑘≢0mod4, (3.34)where 𝑣₁,𝑣₂∈𝑁.

Proof. Firstly, let us consider the 3-generator case. We first note that |𝑥|=4, |𝑦|=6, and |𝑧|=6.(i)If 𝑘=2,𝑃₂(⟨2,3,3⟩;𝑦,𝑧,𝑥)=48 and 𝐵𝑃₂(⟨2,3,3⟩;𝑦,𝑧,𝑥)=12 since 𝑥𝜃=𝑦²𝑥𝑦, 𝑦𝜃=𝑥𝑧⁴𝑥, and 𝑧𝜃=𝑦²𝑥𝑦² where 𝜃 is an outer automorphism of order 4. (ii)If 𝑘>2,let 𝑘≡0mod4, then 𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦,𝑧)=3𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦,𝑧) and 𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦,𝑧)=𝑣₁ℎ_𝑘(6) since 𝑥𝜃=𝑦𝑥𝑦⁵, 𝑦𝜃=𝑧³𝑥𝑦, and 𝑧𝜃=𝑥𝑦²𝑥 where 𝜃 is an inner automorphism induced by conjugation by 𝑧³𝑦𝑥;
let 𝑘≢0mod4, then 𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦,𝑧)=𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦,𝑧) and 𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦,𝑧)=𝑣₂ℎ_𝑘(6) since 𝑥𝜃=𝑥, 𝑦𝜃=𝑦, and 𝑧𝜃=𝑧 where 𝜃 is an inner automorphism induced by conjugation by 𝐱^𝟐.
The proofs are similar to the proofs of Theorems 3.5.(i) and 3.5.(ii) and are omitted.
Secondly, let us consider the 2-generator case. We first note that |𝑥|=4, |𝑦|=6, and |𝑥𝑦|=6.
(i′) If 𝑘=2,𝑃₂(⟨2,3,3⟩;𝑥,𝑦)=48 and 𝐵𝑃₂(⟨2,3,3⟩;𝑥,𝑦)=12 since 𝑥𝜃=𝑦𝑥𝑦² and 𝑦𝜃=𝑦²𝑥 where 𝜃 is an outer automorphism of order 4. (ii′) If 𝑘>2, let 𝑘≡0mod4, then 𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦)=3𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦) and 𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦)=𝑣₁ℎ_𝑘(6) since 𝑥𝜃=𝑦⁵𝑥𝑦, 𝑦𝜃=𝑦𝑥, and 𝑧𝜃=𝑥𝑦²𝑥 where 𝜃 is an inner automorphism induced by conjugation by 𝑦⁵𝑥,
let 𝑘≢0mod4, then 𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦)=𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦) and 𝐵𝑃_𝑘(⟨2,3,3⟩;𝑥,𝑦)=𝑣₂ℎ_𝑘(6) since 𝑥𝜃=𝑥 and 𝑦𝜃=𝑦 where 𝜃 is an inner automorphism induced by conjugation by 𝑥².
The proofs are similar to the proofs of Theorem 3.6.(i′) and Theorem 3.6.(ii′) and are omitted.

Theorem 3.9. The periods of the k-nacci sequences are 𝑘+1, and the basic period of the basic k-nacci sequences is 𝑘+1 in 𝐷₂  four-group.

Proof. We have the presentation 𝐷₂=⟨𝑥,𝑦∶𝑥²=𝑦²=𝑒,𝑥𝑦=𝑦𝑥⟩. 𝑃_𝑘(𝐷₂;𝑥,𝑦)=𝑘+1; see [14] for a proof and 𝐵𝑃_𝑘(𝐷₂;𝑥,𝑦)=𝑘+1 since 𝑥𝜃=𝑥 and 𝑦𝜃=𝑦 where 𝜃 is an inner automorphism induced by conjugation by 𝑥.

Acknowledgments

The authors thank the referees for their valuable suggestions which improved the presentation of the paper. This paper was supported by the Commission for the Scientific Research Projects of Kafkas Univertsity, Project no. 2010-FEF-61.