K-nacci Sequences in Finite Triangle Groups

A k-nacci sequence in a finite group is a sequence of group elements x0, x1, x2, . . . , xn, . . . for which, given an initial seed set x0, x1, x2, . . . , xj−1 , each element is defined by xn x0x1 . . . xn−1, for j ≤ n < k, and xn xn−kxn−k 1 . . . xn−1, for n ≥ k. We also require that the initial elements of the sequence, x0, x1, x2, . . . , xj−1, generate the group, thus forcing the k-nacci sequence to reflect the structure of the group. The K-nacci sequence of a group generated by x0, x1, x2, . . . , xj−1 is denoted by Fk G;x0, x1, . . . , xj−1 and its period is denoted by Pk G;x0, x1, . . . , xj−1 . In this paper, we obtain the period of K-nacci sequences in finite polyhedral groups and the extended triangle groups.


Introduction
The Fibonacci sequences and their related higher-order tribonacci, quatranacci, k-nacci are generally viewed as sequences of integers.In 1 the Fibonacci length of a 2-generator group is defined, thus extending the idea of forming a sequence of group elements based on a Fibonacci-like recurrence relation first introduced by Wall in 2 .There he considered the Fibonacci length of the cyclic group C n .The concept of Fibonacci length for more than two generators has also been considered, see, for example 3, 4 .Also, the theory has been expanded to the nilpotent groups, see, for example 5-7 .Other works on Fibonacci length are discussed in, for example, 8-12 .Knox proved that the periods of k-nacci k-step Fibonacci sequences in dihedral groups are equal to 2k 2 13 .Campbell and Campbel, examined the behaviour of the Fibonacci length of the finite polyhedral, binary polyhedral groups, and related groups in 14 .This paper discusses the period of k-nacci Fibonacci sequences in the polyhedral groups 2, 2, 2 , n, 2, 2 , 2, n, 2 , 2, 2, n for any n and in the extended triangle groups E 2, 2, 2 , E n, 2, 2 , E 2, n, 2 , E 2, 2, n for any n > 2. We consider polyhedral groups both as 2-generator and as 3-generator groups.A 2-step Fibonacci sequence in the integers modulo m can be written as F 2 Z m ; 0, 1 .A 2-step Fibonacci sequence of group elements is called a Fibonacci sequence of a finite group.A finite group G is k-nacci sequenceable if there exists a knacci sequence of G such that every element of the group appears in the sequence.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 x 0 , x 1 , x 2 , x 3 , x 4 , x 1 , x 2 , x 3 , x 4 , x 1 , x 2 , x 3 , x 4 , . . . is periodic after the initial element x 0 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 x 0 , x 1 , x 2 , x 3 , x 4 , x 0 , x 1 , x 2 , x 3 , x 4 , . . . is simply periodic with period 5.It is important to note that the Fibonacci length depends on the chosen generating ntuple for a group.

Definition 1.1. For a finitely generated group G
A where A {a 1 , a 2 , . . ., a n } the sequence Notice that the orbit of a k-generated group is a k-nacci sequence.The orbits of n, 2, 2 , 2, n, 2 , 2, 2, n for any n > 2 and E 2, q, 2 for any q > 2 are studied in 14 .Proof.Firstly, let us consider the 2-generator case.Notice that G 2 is Z 2 ⊕ Z 2 and P k Z 2 ; 0, 1 k 1.Under these identifications, since the period of a Fibonacci sequence in a direct product of groups is the least common multiple of the periods in each the factors we get P k G 2 ; x, y k 1.On the other hand, since z xy the formulas in the "three generator case" with recurrences of period k 1 are the same as the formulas the two generator case as long as k ≥ 4.

2.6
Thus, using the above information the sequence reduces to where x j e for 3 ≤ j ≤ k − 1.Thus, x i e, . . ., e.

2.8
It follows that x k j e for 4 ≤ j ≤ k.We also have, x i z.

2.9
Since the elements succeeding x 2k 2 , x 2k 3 , x 2k 4 , depend on x, y, and z for their values, the cycle begins again with the 2k 2nd element; that is, 1 If there is no t ∈ 3, k − 2 such that t is a odd factor of n, then

2.11
2 Let α be the biggest odd factor of n in 3, k − 2 .Then two cases occur: ii if β is the biggest odd number which is in 3, k − 2 and β α3 j for j ∈ N, then

2.13
Proof.We consider G n as D n , the dihedral group of 2n elements.Now D n being the group of symmetries of the regular polygon with n elements admits a presentation as the group generated by the two matrices: Under these identifications, we can take z i If k 2, we have the sequence xy, x 6 y, x 7 x, x 8 z, . . . .

2.15
Thus we get P 2 G n ; y, x, z 6.

2.17
The 4-nacci sequence can be said to form layers of length 10.Using the above, the 4-nacci sequence becomes  iii If k ≥ 5, the first k 1 elements of the sequence are x 0 x, x 1 y, x 2 z, x 3 z n , x 4 z 2n , . . ., x k z 2 k−3 n .

2.19
Thus, using the above information, the sequence reduces x 0 x, x 1 y, x 2 z, x 3 e, x 4 e, . . ., e, 2.20 where x j e for 3 ≤ j ≤ k.Now we consider what happens to the k-nacci sequence when we have a section of the form . . ., z τ x, zx, z, . . .: x i z u k−4 , . . . .

2.21
The k-nacci sequence can be said to form layers of length 2k 2 .Using the above, the k-nacci sequence becomes x 0 x, x 1 y, x 2 z, x 3 e, . . ., x k z 2 k−3 n e, . . .,

2.22
So, we need the smallest i ∈ N such that 4i nv 1 and 8i 2 4i nv 2 for v 1 , v 2 ∈ N.
1 If there is no t ∈ 3, k − 2 such that t is an odd factor of n, there are 3 subcases.
2 Let α odd be the biggest factor of n in 3, k − 2 .Then two cases occur: i If α3 j / ∈ 3, k − 2 for j ∈ N, then there are 3 subcases.ii' If β is the biggest odd number which is in 3, k − 2 and β α3 j for j ∈ N, then there are 3 subcases.This completes the proof.
In the case of 2-generator the group has the presentation x, y : x 2 y 2 xy n e and the period is the same as in the 3-generator case and proof is similar.

3.1
The extended triangle groups are a very important class of groups closely linked to automorphism groups of regular maps, see 17 .The triangle groups (polyhedral groups), p, q, r are index two subgroups of extended triangle groups.To see this, let X xy, Y yz and Z zx in E p, q, r and then use the obvious epimorphism.We get the following three cases for E p, q, r : 1 the Euclidean case if 1/p 1/q 1/r 1, 2 the elliptic case if 1/p 1/q 1/r > 1, 3 the hyperbolic case if 1/p 1/q 1/r < 1.
The group E p, q, r is finite if and only if 1/p 1/q 1/r > 1.
For more information on these groups, see 14, 18 .
Theorem 3.2.Let E 2 be the group defined by the presentation x, y, z : x 2 y 2 z 2 xy 2 yz 2 zx 2 e .Then P k E 2 ; x, y, z k 1 for k > 2.

3.5
Proof.Since y has order 2 and commutes with x and z it follows that E n Z 2 ⊕ D n .As a group of matrices, the can be identified with a group of 3 × 3 matrices of form Now, since the period of a Fibonacci sequence in a direct product of groups is the least common multiple of the periods in each the factors and from a similar argument applied to Theorem 2.4 the proof is done.

3 . 3 2ii
Let α be the biggest odd factor of n in 3, k − 3 .Then two cases occur:i if α3 j / ∈ 3, k − 3 for j ∈ N, then P k E n ;x, y, z if β is be the biggest odd number which is in 3, k − 3 and β α3 j for j ∈ N, then P k E r ; x, y, z
Proof.Let us consider the 3-generator case.We first note that the orders of x, y, and z are n, 2, 2, respectively.If k 2, we have the sequence x, y, z, yz, zyz, z, x, y, . . ., 2.4 which has period 6.If k 3, we have the sequence x, y, z, xyz e, yz, zyz, z, e, x, y, z, . . ., 2.5 which has period 8.If k ≥ 4, the first k elements of sequence are x 2 x 2k 4 , . . . .∼ 2, n, 2 ∼ 2, 2, n ∼ D n for any n > 2 and using Tietze transformations we can obtain the same presentation for this groups, it is easy to show that for 2-generator P k G Let G n , n > 2, be the group defined by the presentation x, y, z : x 2 y 2 z n n ; x, y 2k 2 in the groups n, 2, 2 , 2, n, 2 , and 2, 2, n .Theorem 2.4.
If there is no t ∈ 3, k − 3 such that t is an odd factor of n, then