The Polytopic-k-Step Fibonacci Sequences in Finite Groups

We study the polytopic-k-step Fibonacci sequences, the polytopic-k-step Fibonacci sequences modulo m, and the polytopic-k-step Fibonacci sequences in finite groups. Also, we examine the periods of the polytopic-k-step Fibonacci sequences in semidihedral group SD2m.


Introduction
The well-known k-step Fibonacci sequence {F k n } k ≥ 2 is defined as for n ≥ 0.

1.1
Let {a j } k−1 j 0 k ≥ 2, a k−1 / 0 be a sequence of real numbers.A k-generalized Fibonacci sequence {V n } ∞ n 0 is defined by the following linear recurrence relation of order k: where V 0 , . . ., V k−1 are specified by the initial conditions.The k-step Fibonacci sequence, the k-generalized Fibonacci sequence, and their properties have been studied by several authors; see, for example, 1-5 .
The k-step Fibonacci sequence is a special case of a sequence which is defined as a linear combination by Kalman as follows where c 0 , c 1 , . . ., c k−1 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 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.
Definition 1.1.For a finitely generated group G A , where A {a 1 , a 2 , . . ., a n }, the sequence is periodic, then the length of the period of the sequence is called the Fibonacci length of G with respect to generating set A, written as LEN A G 7 .
Definition 1.2.For every integer k, where 2 ≤ k ≤ LEN A G , the sequence {y i } ∞ 1 of the elements of G defined by is called a k-step generalized Fibonacci sequence of G, for some positive integers α 1 , α 2 , . . ., α k 8 .
Definition 1.3.A k-nacci sequence in a finite group is a sequence of group elements x 0 , x 1 , x 2 , x 3 , . . ., x n , . . .for which, given an initial seed set x 0 , . . ., x j−1 , each element is defined by We also require that the initial elements of the sequence, x 0 , . . ., 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 seeded by x 0 , . . ., x j−1 is denoted by F k G; x 0 , . . ., x j−1 and its period is denoted by P k G; x 0 , . . ., x j−1 9 .
The Fibonacci sequence, the k-nacci sequence, and the generalized order-k 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, 9-16 .Formulas which classified according to certain rules for this periods are critical to be used in cryptography, see, for example, 17-19 .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 k-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-k-step Fibonacci sequence that exponent of n tnd term is determined that α k−t−1 k−t formula to finite groups and named the polytopic-k-step Fibonacci sequence in finite groups as polytopic-k-nacci sequence.Because of varying both α and according to the number of step and the exponent of each term of this is determined according to a certain rule, the polytopic-k-step Fibonacci sequence is more useful and more general than the k-nacci sequences and the generalized order-k Pell sequence which varying only by the number of step.So that considered by different α 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-k-step Fibonacci sequence to finite groups, more useful and more general formulas than formulas used to obtain periods of the k-nacci and the generalized order-k Pell sequence in finite groups are obtained to be used in cryptography.
In this paper, the usual notation p is used for a prime number.

The Polytopic-k-Step Fibonacci Sequences
The well-known k-topic numbers are defined as When k 2, the k-topic numbers, P k n , are reduced to the triangular numbers.In 20 , Gandhi and Reddy obtained triangular numbers in the generalized Pell sequence {P α n } and generalized associated Pell sequence {Q α n } which are defined for a fixed α > 0, respectively, as for n ≥ 0.

2.2
Now we define for a fixed integer α > 0, a new sequence called the polytopic-k-step Fibonacci sequence {F for n ≥ 0.

2.3
Obviously, if we take α 1 in 2.3 , then this sequence reduces to the well-known k-step Fibonacci sequence.When α ≥ 2 and k 2 in 2.3 , we call {F 2,α n } the polytopic Fibonacci sequence.
By 2.3 , we can write The matrix M is called the polytopic-k-step Fibonacci matrix.
We obtain that the polytopic Fibonacci sequences {F

2,α n
} are generated by a matrix Q α for a fixed integer α ≥ 2: which can be proved by mathematical induction.

The Polytopic-k-Step Fibonacci Sequences Modulo m
In this section we examine the polytopic-k-step Fibonacci sequences modulo m for α ≥ 2 and k ≥ 2.
Reducing the polytopic-k-step Fibonacci sequence by a modulus m, we can get a repeating sequence denoted by m is the period of the polytopic Fibonacci sequence modulo m.Example 3.2.We have {F 3,4  where by p ¹ α k−1 k we mean that α k−1 k is not divided by p and T the transpose of a matrix.It is clear that 3.2 We then obtain that h α k m is least positive integer h α such that n. Then we have The elements of the matrix M n are in the following forms: We thus obtain that

3.6
So we get that M n ≡ I mod p a , which yields that n is divisible by | M p a |.We are done.Proof.Let q be a positive integer.Since M h α k p q 1 ≡ I mod p q 1 , that is, M h α k p q 1 ≡ I mod p q , we get that h α k p q 1 is divided by h α k p q .On the other hand, writing M h α k p q I a q ij p q , we have M h α k p q p I a q ij p q p p i 0 p i a q ij p q i ≡ I mod p q 1 , 3.7 which yields that h α k p q p is divided by h α k p q 1 .Therefore, Table 1 list some primes for which the conjecture is true when k 5 and α 5.

The Polytopic-k-Nacci Sequences in Finite Groups
Definition 4.1.For a finitely generated group G A , where A {a 1 , a 2 , . . ., a n }, we define the polytopic Fibonacci orbit F α A G with respect to the generating set A to be the sequence {x i } of the elements of G such that Definition 4.3.A polytopic-k-nacci sequence in a finite group is a sequence of group elements x 0 , x 1 , . . .x n , . . .for which, given an initial seed set x 0 , . . ., x j−1 , each element is defined by

4.3
It is required that the initial elements of the sequence, x 0 , . . ., x j−1 , generate the group, thus, forcing the polytopic-k-nacci sequence to reflect the structure of the group.We denote the polytopic-k-nacci sequence of a group G generated by x 0 , . . ., x j−1 by F α k G; x 0 , . . ., x j−1 .

4.4
It is important to note that the polytopic Fibonacci orbit of a k-generated group is a polytopick-nacci sequence.
The classic polytopic Fibonacci sequence in the integers modulo m can be written as F α 2 m ; 0, 1 .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-k-nacci sequence in a finite group is periodic.
Proof.The proof is similar to the proof of Theorem 1 in 6 and is omitted.
From the definition, it is clear that the period of a polytopic-k-nacci sequence in a finite group depends on the chosen generating set and the order in which the assignments of x 0 , x 1 , . . .x n−1 are made.Definition 4.6.Let G be a finite group.If there exists a polytopic-k-nacci sequence of the group G such that every element of the group G appears in the sequence, then the group G is called polytopic-k-nacci sequenceable.
It is important to note that the direct product of polytopic-k-nacci sequenceable groups is not necessarily polytopic-k-nacci sequenceable.Consider that the group C 2 × C 4 is defined by the presentation x, y | x 2 y 4 e, xy yx .

4.5
The polytopic Fibonacci sequences of the group C 2 × C 4 for α 2 are for every m ≥ 4. Note that the orders a and b are 2 m−1 and 2, respectively.
Theorem 4.7.The periods of the polytopic-k-nacci sequences in the group SD 2 m for initial (seed) set, a, b, and α 2 are as follows: Proof.i If k 2, we have the polytopic-2-nacci sequence for α 2 :

4.10
By mathematical induction, it is easy to prove that

4.11
So we get By mathematical induction, it is easy to prove that 3 x 0 a, x 1 b, x 2 a 3 , x 3 a 2 m−1 −2 b, x 4 a 14 , x 5 a u   Proof.The proof is similar to the proof of Theorem 4.5 and is omitted.
mod m .It has the same recurrence relation as in 2.3 .Theorem 3.1.{F k,α m } is a periodic sequence for k ≥ 2 and α ≥ 2.

2 2 2 2 2 2 m− 2 x 0 and x h 2 2 2 m−2 1 x 1 . 2 2 2 m− 2 .
depend on a and b for their values, the cycle begins again with the h m−2 nd, that is, x h Thus, the period of F 2 2 SD 2 m ; a, b is h If k 3, we have the polytopic-3-nacci sequence for α 2:
with a ij 's being integers, A mod m means that every entry of A is reduced modulo m, that is, A mod m a ij mod m .Let M p a {M i mod p a | i ≥ 0} be a cyclic group, and let | M p a | denote the order of M p a with p ¹ α k−1 k ij and the latter holds if, and only if, there is an a Conjecture 3.5.Let α ≥ 2. If p ≥ k, then there exists a σ with 0