Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 431840, 12 pages
doi:10.1155/2011/431840
Research Article
The Polytopic-k-Step Fibonacci Sequences in
Finite Groups
Ömür Deveci
Department of Mathematics, Faculty of Art and Science, Kafkas University,
36100 Kars, Turkey
Correspondence should be addressed to Ömür Deveci, odeveci36@yahoo.com.tr
Received 22 May 2011; Revised 26 July 2011; Accepted 27 July 2011
Academic Editor: Binggen Zhang
Copyright q 2011 Ömür Deveci. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 .
1. Introduction
The well- known k-step Fibonacci sequence {Fnk } k ≥ 2 is defined as
k
k
F0 0, . . . , Fk−2 0,
k
k
k
k
Fk−1 1,
k
Fnk Fnk−1 Fnk−2 · · · Fn
for n ≥ 0.
1.1
Let {aj }k−1
k ≥ 2, ak−1 / 0 be a sequence of real numbers. A k-generalized Fibonacci sej0
quence {Vn }∞
n0 is defined by the following linear recurrence relation of order k:
Vn1 a0 Vn a1 Vn−1 · · · ak−1 Vn−k−1 ,
for n ≥ k − 1,
1.2
where V0 , . . . , Vk−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.
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Discrete Dynamics in Nature and Society
The k-step Fibonacci sequence is a special case of a sequence which is defined as a
linear combination by Kalman as follows
ank c0 an c1 an1 · · · ck−1 ank−1 ,
1.3
where c0 , c1 , . . . , ck−1 are real constants. In 6, Kalman derived a number of closed-form
formulas for the generalized sequence by companion matrix method as follows:
Ak aij k×k
⎡
c0
⎢
⎢1
⎢
⎢
⎢0
⎢
⎢
⎢0
⎢
⎢
⎢.
⎢ ..
⎣
0
c1 c2 · · · ck−2 ck−1
0 0 ···
0
1 0 ···
0
0 1 ···
0
..
.
..
.
..
.
0 0 ···
1
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥.
0 ⎥
⎥
⎥
.. ⎥
. ⎥
⎦
0
1.4
Then, by an inductive argument he obtained
⎡
a0
⎤
⎡
an
⎤
⎢
⎥ ⎢
⎥
⎢ a1 ⎥ ⎢ an1 ⎥
⎢
⎥
⎥
⎢
⎥ ⎢
⎥
Ank ⎢
⎢ .. ⎥ ⎢ .. ⎥.
⎢ . ⎥ ⎢ . ⎥
⎣
⎦ ⎣
⎦
ak−1
ank−1
1.5
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 {a1 , a2 , . . . , an }, the
sequence xi ai1 , 0 ≤ i ≤ n − 1, xin nj1 xij−1 , i ≥ 0, is called the Fibonacci orbit of G with
respect to the generating set A, denoted by FA G. If FA G 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 LENA G 7.
Definition 1.2. For every integer k, where 2 ≤ k ≤ LENA G, the sequence {yi }∞
1 of the
elements of G defined by
yi xi , i 1, . . . , k,
α1 α
α
yi−k1 2 · · · yi−1 k ,
yi yi−k
i≥k1
1.6
Discrete Dynamics in Nature and Society
3
is called a k-step generalized Fibonacci sequence of G, for some positive integers α1 , α2 , . . . , αk
8.
Definition 1.3. A k-nacci sequence in a finite group is a sequence of group elements x0 ,
x1 , x2 , x3 , . . . , xn , . . . for which, given an initial seed set x0 , . . . , xj−1 , each element is defined
by
xn ⎧
⎨x0 x1 · · · xn−1
for j ≤ n < k,
⎩x x
n−k n−k1 · · · xn−1
for n ≥ k.
1.7
We also require that the initial elements of the sequence, x0 , . . . , 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 G seeded by x0 , . . . , xj−1 is denoted by Fk G; x0 , . . . , xj−1 and its period is denoted by
Pk G; x0 , . . . , xj−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
Fibonacci sequence that
the polytopic-k-step
formula
to
finite
groups and named the
exponent of n tnd term is determined that αk−t−1
k−t
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.
2. The Polytopic-k-Step Fibonacci Sequences
The well-known k-topic numbers are defined as
nn 1n 2 · · · n r − 1
Pk n k!
nk−1
k
.
2.1
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Discrete Dynamics in Nature and Society
When k 2, the k-topic numbers, Pk n, are reduced to the triangular numbers. In 20,
α
Gandhi and Reddy obtained triangular numbers in the generalized Pell sequence {Pn } and
α
generalized associated Pell sequence {Qn } which are defined for a fixed α > 0, respectively,
as
α
P0
α
0,
α
Q0
P1
α
Q1
α
1,
α
Qn2
1,
α
Pn2 α 1Pn1 α α
1Qn1
αα 1 α
Pn
2
αα 1 α
Qn
2
for n ≥ 0,
2.2
for n ≥ 0.
Now we define for a fixed integer α > 0, a new sequence called the polytopic-k-step Fibonacci
k,α
sequence {Fn }, by
k,α
k,α
Fnk
k,α
k,α
0, . . . , Fk−2 0,
Fk−1 1,
F0
α1
αk−2
αk−1
k,α
k,α
k,α
k,α
αFnk−1 Fnk−2 · · · Fn1 Fn
2
k−1
k
for n ≥ 0.
2.3
Obviously, if we take α 1 in 2.3, then this sequence reduces to the well-known k-step
2,α
Fibonacci sequence. When α ≥ 2 and k 2 in 2.3, we call {Fn } the polytopic Fibonacci
sequence.
By 2.3, we can write
⎡
k,α
Fnk
⎢
⎢ F k,α
⎢ nk−1
⎢
⎢ k,α
⎢ Fnk−2
⎢
⎢
..
⎢
⎢
.
⎣
k,α
Fn1
⎤
⎡
α
⎥ ⎢
⎥ ⎢
⎥ ⎢
1
⎥ ⎢
⎥ ⎢
⎥⎢
0
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎢
⎥ ⎢ ..
⎦ ⎣.
α1
0
2
0
1
..
.
0
⎤⎡
k,α
αk−2
αk−1
F
···
⎥⎢ nk−1
⎥⎢ k,α
k−1
k
⎥⎢ F
⎥⎢ nk−2
···
0
0
⎥⎢ k,α
⎥⎢ F
⎥⎢ nk−3
···
0
0
⎥⎢
⎥⎢
..
⎥⎢
..
..
.
⎥⎣
.
.
⎦
k,α
Fn
···
1
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
2.4
for the polytopic-k-step Fibonacci sequence. Let
M mij k×k
⎤
⎡ α1
αk−2
αk−1
···
⎢α
⎥
⎢
⎥
2
k−1
k
⎢
⎥
⎢1
⎥
0
···
0
0
⎢
⎥
⎢
⎥
⎢0
⎥.
1
·
·
·
0
0
⎢
⎥
⎢
⎥
⎢.
⎥
.
.
.
..
..
..
⎢ ..
⎥
⎣
⎦
0
0
···
1
The matrix M is called the polytopic-k-step Fibonacci matrix.
0
2.5
Discrete Dynamics in Nature and Society
5
2,α
We obtain that the polytopic Fibonacci sequences {Fn
for a fixed integer α ≥ 2:
} are generated by a matrix Qα
⎡
⎤
2,α αα 1 2,α
F
F
n
n1
⎢
⎥
2
Qα n ⎣
⎦
2,α αα 1 2,α
Fn−1
Fn
2
⎡
⎤
αα 1
α
⎦,
Qα ⎣
2
1
0
2.6
which can be proved by mathematical induction.
3. 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
k,α
k,α
k,α
k,α
F k,α m F0 m, F1 m, F2 m, . . . , Fi m, . . . ,
k,α
where Fi
k,α
m Fi
3.1
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.
Proof. Let Uk {x1 , x2 , . . . , xk | 0 ≤ xi ≤ m − 1}. Then we have that |Uk | mk is finite,
k,α
k,α
k,α
that is, for any a ≥ 0, there exist b ≥ a such that Fa1 m ≡ Fb1 m, . . . , Fak m ≡
k,α
k,α
Fbk m. From the definition of the polytopic-k-step Fibonacci sequence {Fn } we have
k,α
k,α αk−1 k,α
k,α
k,α
k,α
k,α
Fnk αFnk−1 α1
Fnk −
Fnk−2 · · · αk−2
Fn1 Fn , that is, αk−1
Fn
2
k−1
k
k
k,α
k,α
k,α
k,α
k,α
Fnk−2 − · · · − αk−2
Fn1 . Then we can easily get that Fa m ≡ Fb m,
αFnk−1 − α1
2
k−1
k,α
k,α
k,α
Fa−1 m ≡ Fb−1 m, . . . , F2
k,α
{Fn
k,α
k,α
m ≡ Fb−a2 m and F1
k,α
m ≡ Fb−a1 m, which implies that
} is a periodic sequence.
α
Let hk m denote the smallest period of {F k,α m}, called the period of the polyα
topic-k-step Fibonacci sequence modulo m. When k 2, h2 m is the period of the polytopic
Fibonacci sequence modulo m.
Example 3.2. We have {F 3,4 3} {0, 0, 1, 1, 2, 2, 0, 0, 1, . . .} and then repeat. So we get
4
h3 3 6.
By elementary number theory it is easy to prove that if m ti1 piei , t ≥ 1, where pi ’s
α
α
are distinct primes, then hk m Icmhk piei .
For a given matrix A aij with aij ’s being integers, A mod m means that every
entry of A is reduced modulo m, that is, A mod m aij mod m. Let Mpa {Mi mod
αk−1 pa | i ≥ 0} be a cyclic group, and let |Mpa | denote the order of Mpa with p
k
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Discrete Dynamics in Nature and Society
αk−1 we mean that αk−1
is not divided by p and T the transpose of a
where by p
k
k
matrix. It is clear that
T
k,α
k,α
k,α
Mi 1, 0, 0, . . . , 0T mod m Fik−1 m, Fik−2 m, . . . , Fi m .
3.2
α
We then obtain that hk m is least positive integer hα such that
α
Mh 1, 0, 0, . . . , 0T
Theorem 3.3. Let α ≥ 2. If p
T
mod m 1, 0, 0, . . . , 0.
3.3
αk−1 α
, then hk pa |Mpa |.
k
α
α
Proof. It is clear that |Mpa | is divisible by hk pa . Then we need only to prove that hk pa α
is divisible by |Mpa |. Let hk pa n. Then we have
⎡
m11 m12 · · · m1k
⎤
⎥
⎢
⎢m21 m22 · · · m2k ⎥
⎥
⎢
⎥.
Mn ⎢
⎢ ..
.
.
..
.. ⎥
⎥
⎢ .
⎦
⎣
mk1 mk2 · · · mkk
3.4
The elements of the matrix Mn are in the following forms:
k,α
k,α
m11 Fnk−1 ,
k,α
k,α
k,α
mii β1 Fnk−2 β2 Fnk−3 · · · βk−1 Fn
k,α
k,α
k,α
m21 Fnk−2 , . . . , mk1 Fn
k,α
mij η1 Fnk−2 η2 Fnk−3 · · · ηk−1 Fn
1,
,
for 2 ≤ i ≤ k, β1 , β2 , . . . , βk−1 ≥ 0,
for i / j, 1 ≤ i ≤ k, 2 ≤ j ≤ k, η1 , η2 , . . . , ηk−1 ≥ 0.
3.5
We thus obtain that
mii ≡ 1 mod pa , for 1 ≤ i ≤ k,
j.
mij ≡ 0 mod pa , for 1 ≤ i, j ≤ k such that i /
3.6
So we get that Mn ≡ I mod pa , which yields that n is divisible by |Mpa |. We are done.
α
α
Theorem 3.4. Let α ≥ 2, and let t be the largest positive integer such that hk p hk pt . Then
α
hk pa α
pa−t hk p
for every a ≥
holds for every a > 1.
α
α
α
t. In particular, if hk p /
hk p2 , then hk pa α
pa−1 hk p
Discrete Dynamics in Nature and Society
7
α
Proof. Let q be a positive integer. Since Mhk
p , we get that
q
q
aij pq ,
α
hk pq1 is divided by
pq1 α
hk pq .
α
≡ I mod pq1 , that is, Mhk
pq1 On the other hand, writing M
α
≡ I mod
hk pq I
we have
M
α
hk pq p
I
q
aij pq
p
p
p i0
α
i
q
aij pq
i
≡ I mod pq1 ,
α
α
3.7
α
which yields that hk pq p is divided by hk pq1 . Therefore, hk pq1 hk pq or
α
q
α
hk pq1 hk pq p, and the latter holds if, and only if, there is an aij which is not
α
α
t1
divisible by p. Since hk pt /
hk pt1 , there is an aij
α
α
hk pt1 /
hk pt2 .
which is not divisible by p, thus,
The proof is finished by induction on t.
Conjecture 3.5. Let α ≥ 2. If p ≥ k, then there exists a σ with 0 ≤ σ ≤ k such that pk1 − pσ is
α
divided by hk p.
Table 1 list some primes for which the conjecture is true when k 5 and α 5.
4. The Polytopic-k-Nacci Sequences in Finite Groups
Definition 4.1. For a finitely generated group G A, where A {a1 , a2 , . . . , an }, we define
α
the polytopic Fibonacci orbit FA
G with respect to the generating set A to be the sequence
{xi } of the elements of G such that
xi ai1 ,
xin xi αn−1
n
xi1 αn−2
n−1
for 0 ≤ i ≤ n − 1,
α1
2
· · · xin−2 xin−1 α ,
for i ≥ 0,
4.1
α
Example 4.2. Let G A, where A {a1 , a2 , a3 }. FA
G is
x0 a1 ,
x1 a2 ,
x2 a3 ,
xi3 xi αα1α2/6 xi1 αα1/2 xi2 α ,
for i ≥ 0.
4.2
Definition 4.3. A polytopic-k-nacci sequence in a finite group is a sequence of group elements
x0 , x1 , . . . xn , . . . for which, given an initial seed set x0 , . . . , xj−1 , each element is defined by
xn ⎧ αn−2
αn−1
⎪
⎪
⎨x0 n x1 n−1 · · · xn−1 α
for j ≤ n < k,
αk−1
⎪
⎪
⎩x k
n−k
for n ≥ k.
αk−2
k−1
xn−k1
α
· · · xn−1 4.3
It is required that the initial elements of the sequence, x0 , . . . , xj−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 x0 , . . . , xj−1 by Fkα G; x0 , . . . , xj−1 .
8
Discrete Dynamics in Nature and Society
5
Table 1: The length of h5 p.
5
h5 p
5
80525
15372
145992
24388
461760
1749132
1661152
2808
205378
4030224
1419912
44264640
13136325
52856154
78804
12914277518098
47685222
16969333200
89206789920
454331269680
529414856880
518758082640
2319528
2601768
4044120
27298090330
262790931413426025
43159140126
132826492154616
p
5
11
13
23
29
31
43
47
53
59
67
73
97
101
223
397
419
523
607
719
821
853
1009
1523
1613
2011
3011
4021
5059
6037
Result
5
h5 p | p6 − p5
5
h5 p | p6 − p
5
h5 p | p6 − 1
5
h5 p | p6 − 1
5
h5 p | p6 − p3
5
h5 p | p6 − p2
5
h5 p | p6 − 1
5
h5 p | p6 − 1
5
h5 p | p6 − p4
5
h5 p | p6 − 1
5
h5 p | p6 − p2
5
h5 p | p6 − p2
5
h5 p | p6 − p2
5
h5 p | p6 − 1
5
h5 p | p6 − 1
5
h5 p | p6 − p4
5
h5 p | p6 − p
5
h5 p | p6 − p3
5
h5 p | p6 − p2
5
h5 p | p6 − 1
5
h5 p | p6 − p2
5
h5 p | p6 − p2
5
h5 p | p6 − 1
5
h5 p | p6 − p4
5
h5 p | p6 − p4
5
h5 p | p6 − p4
5
h5 p | p6 − p3
5
h5 p | p6 − p
5
h5 p | p6 − p3
5
h5 p | p6 − p2
Example 4.4. Let G A, where A {a1 , a2 , a3 }. F4α G; a1 , a2 , a3 is
x0 a1 ,
x1 a2 ,
x2 a3 ,
x3 x0 αα1α2/6 x1 αα1/2 x2 α ,
xi4 xi αα1α2α3/24 xi1 αα1α2/6 xi2 αα1/2 xi3 α
for i ≥ 0.
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
F2α m; 0, 1. We call a polytopic-2-nacci sequence of a group of elements a polytopic Fibonacci
sequence of a finite group.
Discrete Dynamics in Nature and Society
9
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.
We denote the period of a polytopic-k-nacci sequence Fkα G; x0 , . . . , xj−1 by
α
Pk G; x0 , . . . , xj−1 . When α 1, Fkα G; x0 , . . . , xj−1 and Pkα G; x0 , . . . , xj−1 are reduced to
Fk G; x0 , . . . , xj−1 and Pk G; x0 , . . . , xj−1 , respectively.
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
x0 , x1 , . . . xn−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 C2 × C4 is defined
by the presentation
x, y | x2 y4 e, xy yx .
4.5
The polytopic Fibonacci sequences of the group C2 × C4 for α 2 are
F22 C2 × C4 ; x, y x, y, xy2 , y3 , x, y, . . . ,
F22 C2 × C4 ; y, x y, x, y3 , xy2 , y, x, . . . .
4.6
Since the elements e, xy, and xy3 do not in either sequences, the group C2 × C4 is not
polytopic-2-nacci sequenceable.
The group x has a polytopic Fibonacci sequence
F22 x; e, x e, x, e, x, . . .
4.7
and hence is polytopic-2-nacci sequenceable. The group y has a polytopic Fibonacci
sequence
F22 y ; e, y e, y, y2 , y3 , e, y, . . .
4.8
and hence is polytopic-2-nacci sequenceable.
We will now address the periods of the polytopic-k-nacci sequences in specific classes
of groups. A group SD2m is semidihedral group of order 2m if
m−1
m−2
SD2m a, b | a2 b2 e, b−1 ab a−12
for every m ≥ 4. Note that the orders a and b are 2m−1 and 2, respectively.
4.9
10
Discrete Dynamics in Nature and Society
Theorem 4.7. The periods of the polytopic-k-nacci sequences in the group SD2m for initial (seed) set,
a, b, and α 2 are as follows:
2
i Pk2 SD2m ; a, b hk 2m−2 , for 2 ≤ k ≤ 4.
2
ii Pk2 SD2m ; a, b hk 2m−1 , for k ≥ 5.
Proof. i If k 2, we have the polytopic-2-nacci sequence for α 2 :
x0 a,
x1 b,
x5 a2
2
2m−3
x2m−2 a3
x2m−2 1 a2
m−2
,
m−2
−1·2·32m−2 −1·2·32
m−2
x4 a3 ,
x3 a2
x2 a3 ,
−1·2·3
b,
4.10
b, . . . ,
−1·2·32m−2 −1·2·32 ···2m−2 −1·2·32
m−3
b, . . . .
By mathematical induction, it is easy to prove that
≡ 1 mod 2m−1 , 2m−2 − 1 · 2 · 3 2m−2 − 1 · 2 · 32
m−3
32
4.11
m−3
· · · 2m−2 − 1 · 2 · 32 ≡ 0 mod 2m−1 .
2m−3
2m−3
So we get x2m−2 a3
a, x2m−2 1 a2 −1·2·32 −1·2·3 ···2 −1·2·3 b b. It is easy to
2 m−2
2
m−3
see that h2 2 2
· h2 2 2m−3 · 2 2m−2 . Since the elements succeeding xh2 2m−2 ,
m−2
m−2
2
m−2
2
2
xh2 2m−2 1 , depend on a and b for their values, the cycle begins again with the h2 2m−2 nd,
2
2
that is, xh2 2m−2 x0 and xh2 2m−2 1 x1 . Thus, the period of F22 SD2m ; a, b is h2 2m−2 .
2
2
If k 3, we have the polytopic-3-nacci sequence for α 2:
x0 a,
x1 b,
m−2
x5 a3·442
2m−3
x2m−2 a3
,
m−2
x2 a3 ,
−1·2·32m−2 −1·2·32
2m−3 −1
x2m−2 1 a3
x3 a42
·432
m−3 −2
b,
−1·2·3
3
x6 a3 , . . . ,
·4···42m−2 −1·2·32m−2 −1·2·32 ···2m−2 −1·2·32
2m−3 1
x2m−2 2 a3
≡ 0 mod2
4
32
m−3 −1
·432
m−3 −2
, 3
2
1
≡ 3 mod2
m−1
. So we get x2m−2
·4···42m−2 −1·2·32m−2 −1·2·32 ···2m−2 −1·2·32
b,
4.12
m−3
m−1
m−3
··· .
By mathematical induction, it is easy to prove that 32
m−3
2
x4 a3 ,
b,
−1
m−3
· 4 32
m−3
2m−3
3
a
32
m−3 1
−2
· 4 ··· a, x2m−2 1
a
b b, x
a
a . It is easy
2
2
to see that h3 2m−2 2m−3 · h3 2 2m−3 · 2 2m−2 . Since the elements succeeding
xh2 2m−2 , xh2 2m−2 1 , xh2 2m−2 2 depend on a, b, and a3 for their values, the cycle begins again
3
2
3
2m−2 2
3
3
with h3 2m−2 nd, that is xh2 2m−2 x0 , xh2 2m−2 1 x1 , and xh2 2m−2 1 x2 . Thus, the period
2
3
3
3
of F32 SD2m ; a, b is h3 2m−2 . The proof for k 4 is similar and is omitted.
Discrete Dynamics in Nature and Society
11
ii If k ≥ 5, we have the polytopic-k-nacci sequence for α 2 :
x0 a,
x1 b,
x5 au1 ,
x4h2 2 a25 ,
−2
k
k
x4h2 26 a
k
xi·4h2 2 a
k
m−1
xi·4h2 23 a2
−2
k
,
b,
k
x4h2 24 a30 ,
b,
k
u2 8·λ2
124·i
x4 a14 ,
b,
x4h2 22 a27 ,
k
x4h2 23 a2
,
−2
x6 au2 , . . . , xk auk−4 ,
m−1
x4h2 25 a
x3 a2
x4h2 21 a8 b,
k
u1 8·λ1
m−1
x2 a3 ,
,...,
xi·4h221 a b,
8·i
k
x4h2 2k auk−4 8·λk−4 , . . . ,
4.13
k
xi.4h2 22 a324·i ,
xi·4h224 a1416·i ,
k
k
xi·4h225 au1 8·i·λ1 ,
k
xi·4h2 26 au2 8·i·λ2 , . . . , xi·4h22k auk−4 8·i·λk−4 , . . . ,
k
k
where λ1 , . . . , λk−4 are natural numbers and u1 , . . . , uk−4 are even natural numbers. So we need
the smallest i ∈ such that 8 · i 2m−1 . If we choose i 2m−4 , we obtain xh2 2m−1 a m−1
x0 , xh2 2m−1 1 b x1 , xh2 2m−1 2 a3 x2 , xh2 2m−1 3 a2
k
k
k
−2
k
b x3 , xh2 2m−1 4 a14 x4 ,
k
2
xh2 2m−1 5 au1 x5 , xh2 2m−1 6 au2 x6 , . . ., xh2 2m−1 k auk−4 xk since 2m−2 · hk 2 k
k
2
2
k
hk 2m−1 . So we get Pk2 SD2m ; a, b hk 2m−1 for k ≥ 5.
Theorem 4.8. The periods of the the polytopic-k-nacci sequences in the group SD2m for initial (seed)
sets b, a, and α 2 are as follows:
2
i Pk2 SD2m ; b, a hk 2m−2 for 2 ≤ k ≤ 3,
2
ii Pk2 SD2m ; b, a hk 2m−1 for k ≥ 4.
Proof. The proof is similar to the proof of Theorem 4.5 and is omitted.
Acknowledgment
This project was supported by the Commission for the Scientific Research Projects of Kafkas
University. The Project no. 2010-FEF-61.
References
1 E. Kiliç and D. Tasci, “On families of bipartite graphs associated with sums of generalized order-k
Fibonacci and Lucas numbers,” Ars Combinatoria, vol. 94, pp. 13–23, 2010.
2 G. Y. Lee, “k-Lucas numbers and associated bipartite graphs,” Linear Algebra and its Applications, vol.
320, no. 1-3, pp. 51–61, 2000.
3 K. Lü and W. Jun, “k-step Fibonacci sequence modulo m,” Utilitas Mathematica, vol. 71, pp. 169–178,
2007.
4 E. F. Miles, “Generalized Fibonacci numbers and associated matrices,” The American Mathematical
Monthly, vol. 67, pp. 745–752, 1960.
12
Discrete Dynamics in Nature and Society
5 M. Mouline and M. Rachidi, “∞-generalized Fibonacci sequences and Markov chains,” The Fibonacci
Quarterly, vol. 38, no. 4, pp. 364–371, 2000.
6 D. Kalman, “Generalized Fibonacci numbers by matrix methods,” The Fibonacci Quarterly, vol. 20, no.
1, pp. 73–76, 1982.
7 C. M. Campbell and P. P. Campbell, “The Fibonacci length of certain centro-polyhedral groups,”
Journal of Applied Mathematics and Computing, vol. 19, no. 1-2, pp. 231–240, 2005.
8 A. Sadeghieh and H. Doostie, “Non-abelian sequenceable groups involving α-covers,” Journal of
Sciences, vol. 20, no. 3, pp. 277–282, 2009.
9 S. W. Knox, “Fibonacci sequences in finite groups,” The Fibonacci Quarterly, vol. 30, no. 2, pp. 116–120,
1992.
10 H. Aydin and R. Dikici, “General Fibonacci sequences in finite groups,” The Fibonacci Quarterly, vol.
36, no. 3, pp. 216–221, 1998.
11 C. M. Campbell, H. Doostie, and E. F. Robertson, “Fibonacci length of generating pairs in groups,”
in Applications of Fibonacci Numbers, G. E. Bergum et al., Ed., vol. 3, pp. 27–35, Kluwer Academic
publisher, Dodrecht, The Netherlands, 1990.
12 O. Deveci and E. Karaduman, “On the basic k-nacci sequences in finite groups,” Discrete Dynamics in
Nature and Society, vol. 34, no. 4, pp. 275–283, 2011.
13 Ö. Deveci and E. Karaduman, “The Pell sequences in finitegroups,” Utilitas Mathematica. In press.
14 E. Karaduman and H. Aydin, “-nacci sequences in some special groups of finite order,” Mathematical
and Computer Modelling, vol. 50, no. 1-2, pp. 53–58, 2009.
15 D. D. Wall, “Fibonacci series modulo m,” The American Mathematical Monthly, vol. 67, pp. 525–532,
1960.
16 H. J. Wilcox, “Fibonacci sequences of period n in groups,” The Fibonacci Quarterly, vol. 24, no. 4, pp.
356–361, 1986.
17 A. S. Fraenkel and S. T. Klein, “Robust universal complete codes for transmission and compression,”
Discrete Applied Mathematics, vol. 64, no. 1, pp. 31–55, 1996.
18 G. R. Kaluge, “Penggunaan fibonacci dan Josephus problem dalam algoritma enkripsi transposisisubstitusi,” Makalah IF 3058 Kriptografi-Sem. II Tahun 2010/2011.
19 D. M. Mandelbaum, “Synchronization of codes by means of Kautz’s Fibonacci encoding,” Institute of
Electrical and Electronics Engineers. Transactions on Information Theory, vol. IT-18, pp. 281–285, 1972.
20 B. K. Gandhi and M. J. Reddy, “Triangular numbers in the generalized associated Pell sequence,”
Indian Journal of Pure and Applied Mathematics, vol. 34, no. 8, pp. 1237–1248, 2003.
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