The k-step [alpha]-generalized Pell-Padovan sequence in Finite Groups

Headnote

Abstract: In this paper, we study the k-step a-generalized Pell-Padovan sequence modulo m. We define the k-step a-generalized Pell-Padovan sequences in a finite group and we examine the periods of these sequences. Also, we obtain the periods of the k-step a-generalized Pell-Padovan sequences in the semidihedral group SDnm.

Keywords: Sequence, period, matrix, group.

(ProQuest: ... denotes formulae omitted)

1 Introduction and Preliminaries

The Pell-Padovan sequence {I1 in)} is defined [12,13] recursively by the equation

P(n+3) = 2P(n+1)+P(n)(1)

for n ≥ 0, where P(0) = P(1) = P(2) = 1.

Kahnan [8] mentioned that these sequences are special cases of a sequence which is defined recursively as a linear combination of the preceding k tenns:

...

where c0,ci, * * * ,ck-1 are real constants. In [8], Kahnan derived a number of closed-fonn fonnulas for the generalized sequence by companion matrix method as follows:

...

Then by an inductive argument he obtained that

...

It is well-known that a sequence 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 the period of the sequence.

The study of the linear recurrence sequences in groups began with the earlier work of Wall [14] where the ordinary Fibonacci sequences in cyclic groups were investigated. Recently, many authors have studied on some special linear recurrence sequences in groups; see for example, [1-7,9-11]. Now we extend the concept to the k-step a-generalized Pell-Padovan sequences. In this paper, the usual notation p is used for a prune number.

2 The k-step α-generalized Pell-Padovan sequence

The k-step α-generalized Pell-Padovan sequence is defined as

...(2)

for n > 0, where ....

When k = 2 and α = 1, this sequence reduces to the usual Pell-Padovan sequence, {P (n)}. By (2), we have

...

for the k-step α-generalized Pell-Padovan sequence. Let

...

By inductive argument we have

...(3)

for n ≥ 0.

3 The k-step α-generalized Pell-Padovan sequence modulo m

Reducing the k-step α-generalized Pell-Padovan sequence modulo m, we can get a repeating sequence, denoted by

...

where ... (mod in) and it has the same recurrence relation as in (2).

Theorem 3.1. ... is a simply periodic sequence.

Proof. Let .... Then we have S(Ak+i) = mk+l being finite (S(Ak+1) mean that the number of the elements of Ak+1 ), that is, for any v > 0, there exist w >v + k such that

...

From definition of the the ¿-step a-generalized Pell-Padovan sequence we have..., which hnplies that this sequence is simply periodic. We denote the smallest period of ....

Example 1. We have = { hP5\2 = 15.

For given a matrix ..., ( with c;/'s being integers, M (mod in) means that every entries of M are reduced modulo m, that is, M (mod m) = [eij (mod m)). Let a be an positive integer and let {M)pC1 = {Ml (modpa)| i > 0}. Then, it is clear that the set {M)pC1 is a cyclic group. Let {M)pC1 denote the order of {M)pa.

Let a be an positive integer, then by (3), it is shown that ....

Theorem 3.2. Let t be the largest positive integer such that .... Then hPkpU =..., for every u > t.

Proof. By ... we see that for each positive integer ..., hence. ..., which means that ... divides ... Thus

...

which yields that ... divides .... So, we can write ..., and the latter holds if, and only if, there is a m'P which is not divisible by p. Since ..., there is an inß 1 ' which is not divisible by p, thus, .... The proof is completed by induction on t.

Theorem 3.3. If m = nUp?, (« > i) where pi s are distinct primes, then ...

Proof. Since ... is the period of ..., the sequence ... repeats only after blocks of length ..., (β is a natural numbers). Also, since ... is the period ..., the sequence ... repeats after hPßm tenns for all values i. Thus, hPß is of the form ß JiPß vi for all values of i, and since any such number gives a period of ...

4 The fc-step a-generalized Pell-Padovan sequence in groups

Definition 4.1. For a generating pair (x;y) 2 G, we define the Pell-Padovan orbit Px;y;y (G) = fxig and co-Pell-Padovan orbit c-Px,y,y (G) = {xi}, respectively as follows:

...

and

...

Definition 4.2. A k-step a-generalized Pell-Padovan sequence in a finite group is a sequence of group elements ao,ai,-an,- for which, given an initial (seed) set ..., each element is defined

...

It is require that the initial elements of the sequence, ao, * * * ,a;-_i, generate the group, thus, forcing the k-step a-generalized Pell-Padovan sequence to reflect the structure of the group. We denote the k-step a-generalized Pell-Padovan sequence of a group generated by x0, ***, xj-1 by PPK (G;x0, * * *, xj-1).

The k-step a-generalized Pell-Padovan sequence in a cyclic group C" of order n can be written as PPK (Cn;x,x, * * *, x).

Theorem 4.1. A k-step a-generalized Pell-Padovan sequence in a finite group is periodic.

Proof. Let G be a finite group and |G| be the order of G. Since there are \G\k 1 distinct ¿+1-tuples of elements of the group G, at least one of the k+ 1-tuples appears twice in a ¿-step a-generalized Pell-Padovan sequence of the group G. Because of the repeating, the ¿-step a-generalized Pell-Padovan sequence is periodic.

We denote the period of the sequence PPk (G;xo, * * *. xi \ ) by Perk (G;xo, * * *. xi \ ). From the definition it is clear that the period of the ¿-step a-generalized Pell-Padovan sequence in a finite group depends on the chosen generating set and the order in which the assignments of x0,x1,x2, * * *, xj-1.

It is clear that ....

A group SD2«> is semidihedral group of order 2m if

...

for every m ≥ 4. Note that the orders x and y are 2m-1 and 2, respectively.

Theorem 4.2. The periods of the ¿-step a-generalized Pell-Padovan sequences in the semidihedral group STL"'for initial (seed) set x. y are as follows:

i. Per2 (SD2m;x, y) = 3-2m-2

ii. Perk (SD2m;x, y) = hP1k2 * 2m-2 for 3 ≤ k ≤ 4.

iii. Per2 (SD2m;x, y) = hP1k2 * 2m-3 for k ≥ 5.

Proof.i. The sequence PP2 (Sl)vnx. y) is

...

Using the relations of the SD2m, this sequence becomes:

...

So we need the smallest such that 2'"~1 - 2i = 0, i is a natural numbers. Thus, we obtain x3.2,"-2 = x, %2»i-2+1 = y and-U.2"'-2+2 = y- Since the elements succeeding x3.2m-2,x2"2m-2+l,x2,.2m-i+2, depend on x,y and y for their values, the cycle begins again with the (3-2"' 2)n<l element. So we get ....

ii. Note that hP132 = 15 and the sequence PP3 (Sl)vxx. y) is

...

So, by the relations of the SD?>», this sequence becomes:

...

So we need the smallest such that 2'"= 4i, i is a natural numbers. If 2"' 3 = i, we obtain ... and *30 2»i-2+3 = ** Since the elements succeeding *iS 2'" 2-*13 2"' 2 i* *|3 2'" 2 2-*132'" 2 3 depend on *,y,y and * for their values, the cycle begins again with the (15 * 2"'-2)nd element. So we get ....

The proof for k 4 is similar to the above and it is omitted.

iii. Let k ≥ 5. We have the sequence

...

So, we need the smallest such that 2'"~1 = 8i, i is a natural numbers. If .... Thus, the cycle begins again with the ... element. So we get ....

5 Acknowledgment

This Project was supported by the Commission for the Scientific Research Projects of Kafkas University. The Project number. 2013-FEF-72.

References

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