K-nacci Sequences in Finite Triangle Groups
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 K-nacci sequence of a group generated by is denoted by and its period is denoted by . In this paper, we obtain the period of K-nacci sequences in finite polyhedral groups and the extended triangle groups.
The Fibonacci sequences and their related higher-order (tribonacci, quatranacci, -nacci) are generally viewed as sequences of integers. In  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 . There he considered the Fibonacci length of the cyclic group . 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 -nacci (-step Fibonacci) sequences in dihedral groups are equal to . Campbell and Campbel, examined the behaviour of the Fibonacci length of the finite polyhedral, binary polyhedral groups, and related groups in .
This paper discusses the period of -nacci Fibonacci sequences in the polyhedral groups , , , for any and in the extended triangle groups , , , for any . We consider polyhedral groups both as 2-generator and as 3-generator groups. A 2-step Fibonacci sequence in the integers modulo can be written as A 2-step Fibonacci sequence of group elements is called a Fibonacci sequence of a finite group. A finite group is -nacci sequenceable if there exists a -nacci sequence of 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 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 5. It is important to note that the Fibonacci length depends on the chosen generating - tuple for a group.
Definition 1.1. For a finitely generated group where the sequence , , , , is called the Fibonacci orbit of with respect to the generating set , denoted .
Notice that the orbit of a k-generated group is a -nacci sequence. The orbits of , , for any and for any are studied in .
2. The Groups , , , and
Definition 2.1. The polyhedral group for is defined by the presentation or The polyhedral group is finite if and only if the number is positive, that is, in the case , , , , . Its order is . Using Tietze transformations, we may show that . For more information on these groups see  and [16, pages 67–68]. The groups considered in Theorems 2.3 and 2.4 are the same group, namely, , the dihedral group of elements, except the generators and are different from one theorem to the other.
Theorem 2.2. Let be the group defined by the presentation . Then .
Proof. Firstly, let us consider the 2-generator case. Notice that is and . 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 . On the other hand, since the formulas in the “three generator case” with recurrences of period are the same as the formulas the two generator case as long as .
Theorem 2.3. Let be the group defined by the presentation . Then
Proof. Let us consider the 3-generator case. We first note that the orders of and are , respectively. If we have the sequence
which has period 6. If we have the sequence
which has period 8. If the first elements of sequence are
Thus, using the above information the sequence reduces to
where . Thus,
It follows that . We also have,
Since the elements succeeding depend on and for their values, the cycle begins again with the element; that is, Thus, .
Similarly, it is easy to show that for 2-generator, in and it can be shown that for .
Because of for any and using Tietze transformations we can obtain the same presentation for this groups, it is easy to show that for 2-generator in the groups and .
Theorem 2.4. Let , be the group defined by the presentation
If there is no such that is a odd factor of then Let be the biggest odd factor of in . Then two cases occur:if for , then if is the biggest odd number which is in and for , then
Proof. We consider as , the dihedral group of elements. Now being the group of symmetries of the regular polygon with elements admits a presentation as the group generated by the two matrices:
Under these identifications, we can take and .(i)If , we have the sequence
Thus we get .(ii)If , we have the sequence
Now we consider what happens to the 4-nacci sequence when we have a section of the form :
The 4-nacci sequence can be said to form layers of length 10. Using the above, the 4-nacci sequence becomes
where and .
So, we need the smallest such that and for .
If , and for .
Thus, and .
If , and for .
Thus, and .
If or , and for .
Thus, and .
(iii)If the first elements of the sequence are Thus, using the above information, the sequence reduces where for .
Now we consider what happens to the -nacci sequence when we have a section of the form
The -nacci sequence can be said to form layers of length . Using the above, the -nacci sequence becomes So, we need the smallest such that and for . If there is no such that is an odd factor of , there are 3 subcases.Case 1. If and , and for . So, we get since (where by we mean that divides ).Case 2. If and , and for . So, we get since .Case 3. If or and , and for . So, we get since (2)Let odd be the biggest factor of in . Then two cases occur:If for then there are 3 subcases.Case 1. If and , and for . So, we get since .Case 2. If and , and for . So, we get since .Case 3. If or and , and for . So, we get since (ii’)If is the biggest odd number which is in and for then there are 3 subcases.Case 1. If and , and for . So, we get since .Case 2. If and , and for . So, we get since .Case 3. If or and , and for . So, we get since .This completes the proof.
In the case of 2-generator the group has the presentation and the period is the same as in the 3-generator case and proof is similar.
3. The Groups , , , and
Definition 3.1. The extended triangle group , for , is defined by the presentation
The extended triangle groups are a very important class of groups closely linked to automorphism groups of regular maps, see . The triangle groups (polyhedral groups), are index two subgroups of extended triangle groups. To see this, let and in and then use the obvious epimorphism. We get the following three cases for :(1)the Euclidean case if ,(2)the elliptic case if ,(3)the hyperbolic case if .The group is finite if and only if .
For more information on these groups, see [14, 18].
Theorem 3.2. Let be the group defined by the presentation . Then for .
Proof. Since can be identified with and with , respectively, from a similar argument applied to Theorem 2.2, we get .
Theorem 3.3. Let , be the group defined by the presentation (i)(ii)let .If there is no such that is an odd factor of then Let be the biggest odd factor of in . Then two cases occur:if for , then if is be the biggest odd number which is in and for , then
Proof. Since has order 2 and commutes with and it follows that . As a group of matrices, the can be identified with a group of matrices of form where is a matrix in dihedral group generated by and shown at (2.14). Here, 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.
The authors thank the referees for their valuable suggestions which improved the presentation of the paper.
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