Research Article The Polytopic-k-Step Fibonacci Sequences in Finite Groups

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1 Discrete Dynamics in Nature and Society Volume 011, Article ID , 1 pages doi:101155/011/ Research Article The Polytopic--Step Fibonacci Sequences in Finite Groups Ömür Deveci Department of Mathematics, Faculty of Art and Science, Kafas University, Kars, Turey Correspondence should be addressed to Ömür Deveci, odeveci36@yahoocomtr Received May 011; Revised 6 July 011; Accepted 7 July 011 Academic Editor: Binggen Zhang Copyright q 011 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 wor is properly cited We study the polytopic--step Fibonacci sequences, the polytopic--step Fibonacci sequences modulo m, and the polytopic--step Fibonacci sequences in finite groups Also, we examine the periods of the polytopic--step Fibonacci sequences in semidihedral group SD m 1 Introduction The well- nown -step Fibonacci sequence {Fn } is defined as F 0 0,,F 0, F 1, 1 F F F F n n 1 n n for n 0 11 Let {a j } 1 j 0,a 1 / 0 be a sequence of real numbers A -generalized Fibonacci sequence {V n } n 0 is defined by the following linear recurrence relation of order : V n 1 a 0 V n a 1 V n 1 a 1 V n 1, for n 1, 1 where V 0,,V 1 are specified by the initial conditions The -step Fibonacci sequence, the -generalized Fibonacci sequence, and their properties have been studied by several authors; see, for example, 1 5

2 Discrete Dynamics in Nature and Society The -step Fibonacci sequence is a special case of a sequence which is defined as a linear combination by Kalman as follows a n c 0 a n c 1 a n 1 c 1 a n 1, 13 where c 0,c 1,,c 1 are real constants In 6, Kalman derived a number of closed-form formulas for the generalized sequence by companion matrix method as follows: c 0 c 1 c c c A [ aij ] Then, by an inductive argument he obtained A n a 0 a 1 a n a n 1 15 a 1 a n 1 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 if the first 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 11 For a finitely generated group G A, wherea {a 1,a,,a n },the sequence x i a i 1,0 i n 1, x i n n j 1 x i j 1, i 0, is called the Fibonacci orbit of G with respect to the generating set A, denoted by F A G IfF A 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 LEN A G 7 Definition 1 For every integer, where LEN A G, thesequence{y i } 1 elements of G defined by y i x i, i 1,,, y i ( ) α1 ( ) α ) α y i yi 1 (y i 1, i 1 of the 16

3 Discrete Dynamics in Nature and Society 3 is called a -step generalized Fibonacci sequence of G, for some positive integers α 1,α,,α 8 Definition 13 A -nacci sequence in a finite group is a sequence of group elements x 0, x 1,x,x 3,,x n,for which, given an initial seed set x 0,,x j 1, each element is defined by x 0 x 1 x n 1 for j n<, x n x n x n 1 x n 1 for n 17 We also require that the initial elements of the sequence, x 0,,x j 1, generate the group, thus forcing the -nacci sequence to reflect the structure of the group The -nacci sequence of a group G seeded by x 0,,x j 1 is denoted by F G; x 0,,x j 1 and its period is denoted by P G; x 0,,x j 1 9 The Fibonacci sequence, the -nacci sequence, and the generalized order- 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, 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 -step Fibonacci sequence which is special type of the generalized Fibonacci sequences with polytopic numbers which are a well-nown family of integers, we conveyed the sequence named the polytopic--step Fibonacci sequence that exponent of n tnd term is determined that ( ) α t 1 t formula to finite groups and named the polytopic--step Fibonacci sequence in finite groups as polytopic--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--step Fibonacci sequence is more useful and more general than the -nacci sequences and the generalized order- 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--step Fibonacci sequence to finite groups, more useful and more general formulas than formulas used to obtain periods of the -nacci and the generalized order- 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--Step Fibonacci Sequences The well-nown -topic numbers are defined as P n n n 1 n n r 1! ( ) n 1 1

4 4 Discrete Dynamics in Nature and Society When, the -topic numbers, P n, are reduced to the triangular numbers In 0, Gandhi and Reddy obtained triangular numbers in the generalized Pell sequence {Pn α } and generalized associated Pell sequence {Qn α } which are defined for a fixed α>0, respectively, as P α 0 0, P α 1 1, P α α n α 1 Pn 1 Q α 0 Q α 1 1, Q α n α 1 Q α n 1 α α 1 P α n for n 0, α α 1 Qn α for n 0 Now we definefor a fixed integer α>0, a new sequence called the polytopic--step Fibonacci sequence {Fn,α },by F,α n F,α 0 0,,F,α 0, F,α ( ) ( α 1 α αf,α n 1 F,α n 1 ) 1 1, F,α n 1 ( α 1 ) F,α n for n 0 3 Obviously, if we tae α 1in 3, then this sequence reduces to the well-nown -step Fibonacci sequence When α and in 3, wecall{fn,α } the polytopic Fibonacci sequence By 3, wecanwrite F,α n F,α n 1 F,α n F,α n 1 ( ) ( ) ( ) α 1 α α 1 α F,α n 1 F,α n F,α n 3 F,α n 4 for the polytopic--step Fibonacci sequence Let ( ) ( ) ( ) α 1 α α 1 α 1 M [ m ij ] The matrix M is called the polytopic--step Fibonacci matrix

5 Discrete Dynamics in Nature and Society 5 We obtain that the polytopic Fibonacci sequences {Fn,α } are generated by a matrix Q α for a fixed integer α : α α 1 Q α α, Q α n 1 0 F,α n 1 F,α n α α 1 α α 1 Fn,α F,α n 1 6 which can be proved by mathematical induction 3 The Polytopic--Step Fibonacci Sequences Modulo m In this section we examine the polytopic--step Fibonacci sequences modulo m for α and Reducing the polytopic--step Fibonacci sequence by a modulus m, we can get a repeating sequence denoted by { } { } F,α m F,α 0 m,f,α 1 m,f,α m,,f,α i m,, 31 where F,α i m F,α mod m It has the same recurrence relation as in 3 i Theorem 31 {F,α m } is a periodic sequence for and α Proof Let U { x 1,x,,x 0 x i m 1} Thenwehavethat U m is finite, that is, for any a 0, there exist b a such that F,α a 1 m F,α b 1 m,,f,α a m F,α b m From the definition of the polytopic--step Fibonacci sequence {F,α n } we have F,α αf,α n n 1 ( ) α 1,α F n ( ) α,α 1 F n 1 ( ) α 1,α F n,thatis, ( ) α 1,α F n F,α n αf,α n 1 ( ) α 1,α F n ( ) α,α 1 F n 1 Then we can easily get that F,α a m F,α m, b F,α a 1 m F,α b 1 m,,f,α m F,α m and F,α b a 1 m F,α m, which implies that b a 1 {F,α n } is a periodic sequence Let h α m denote the smallest period of {F,α m }, called the period of the polytopic--step Fibonacci sequence modulo mwhen, h α m is the period of the polytopic Fibonacci sequence modulo m Example 3 We have {F 3,4 3 } {0, 0, 1, 1,,, 0, 0, 1,} and then repeat So we get h By elementary number theory it is easy to prove that if m t i 1 pe i i, t 1,wherep i s are distinct primes, then h α m Icm h α pe i i For a given matrix A a ij with a ij s being integers, A mod m means that every entry of A is reduced modulo m, thatis,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 - ( α 1

6 6 Discrete Dynamics in Nature and Society ( where by p - α 1 ) ( we mean that α 1 ) is not divided by p and T the transpose of a matrix It is clear that (M i 1, 0, 0,,0 T) T [ ] mod m F,α i 1 m,f,α i m,,f,α i m 3 We then obtain that h α m is least positive integer h α such that ( M h α 1, 0, 0,,0 T) T mod m 1, 0, 0,,0 33 Theorem 33 Let α Ifp - ( α 1 ) α,thenh pa M p a Proof It is clear that M p a is divisible by h α pa Then we need only to prove that h α pa is divisible by M p a Leth α pa n Thenwehave m 11 m 1 m 1 m 1 m m M n 34 m 1 m m The elements of the matrix M n are in the following forms: m 11 F,α n 1, m 1 F,α n,,m 1 F,α n, m ii β 1 F,α n β F,α n 3 β 1F,α n 1, for i, β 1,β,,β 1 0, m ij η 1 F,α n η F,α n 3 η 1F,α n for i / j, 1 i, j, η 1,η,,η We thus obtain that m ii 1 ( mod p a), for 1 i, m ij 0 ( mod p a), for 1 i, j such that i / j 36 So we get that M n I mod p a, which yields that n is divisible by M p a Wearedone Theorem 34 Let α, andlett be the largest positive integer such that h α p h α pt Then h α pa p a t h α p for every a t In particular, if h α p / h α p,thenh α pa p a 1 h α p holds for every a>1

7 Discrete Dynamics in Nature and Society 7 Proof Let q be a positive integer Since M h α p q,wegetthath α a q ij p q,wehave pq 1 is divided by h α ( ( M h α pq p I a q ij p q)) p p i 0 pq 1 I mod p q 1,thatis,M h α pq 1 I mod pq On the other hand, writing M h α pq I ( p i ) ( a q ij p q) i I (mod p q 1), 37 which yields that h α pq p is divided by h α pq 1 Therefore, h α pq 1 h α pq or pq 1 h α pq p, and the latter holds if, and only if, there is an a q ij which is not divisible by p Sinceh α pt / h α pt 1,thereisana t 1 ij which is not divisible by p, thus, h α h α pt 1 / h α pt The proof is finished by induction on t Conjecture 35 Let α Ifp, then there exists a σ with 0 σ such that p 1 p σ is divided by h α p Table 1 list some primes for which the conjecture is true when 5andα 5 4 The Polytopic--Nacci Sequences in Finite Groups Definition 41 For a finitely generated group G A, wherea {a 1,a,,a n },wedefine 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 x i n x i ( α n 1 n x i a i 1, for 0 i n 1, ) x i 1 ( α n n 1 ) x i n ( α 1 ) x i n 1 α, for i 0, 41 Example 4 Let G A,whereA {a 1,a,a 3 } F α A G is x 0 a 1, x 1 a, x a 3, x i 3 x i α α 1 α/6 x i 1 α α 1 / x i α, for i 0 4 Definition 43 A polytopic--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 x x n x ( ) ( ) α n 1 α n n n 1 0 x ( ) ( ) α 1 α 1 x n 1 x n 1 α for j n<, n 1 x n 1 α for n 43 It is required that the initial elements of the sequence, x 0,,x j 1, generate the group, thus, forcing the polytopic--nacci sequence to reflect the structure of the group We denote the polytopic--nacci sequence of a group G generated by x 0,,x j 1 by F α G; x 0,,x j 1

8 8 Discrete Dynamics in Nature and Society Table 1: The length of h 5 5 p p h 5 5 p Result 5 5 h 5 5 p p6 p h 5 5 p p6 p h h h 5 5 p p6 p h 5 5 p p6 p h h h 5 5 p p6 p h h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h h h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h h 5 5 p p6 p h 5 5 p p6 p h h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p h 5 5 p p6 p Example 44 Let G A,whereA {a 1,a,a 3 } F α 4 G; a 1,a,a 3 is x 0 a 1, x 1 a, x a 3, x 3 x 0 α α 1 α/6 x 1 α α 1 / x α, x i 4 x i α α 1 α α 3 /4 x i 1 α α 1 α/6 x i α α 1 / x i 3 α for i 0 44 It is important to note that the polytopic Fibonacci orbit of a -generated group is a polytopic-nacci sequence The classic polytopic Fibonacci sequence in the integers modulo m can be written as F α Z m;0, 1 We call a polytopic--nacci sequence of a group of elements a polytopic Fibonacci sequence of a finite group

9 Discrete Dynamics in Nature and Society 9 Theorem 45 A polytopic--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--nacci sequence F α G; x 0,,x j 1 by P α G; x 0,,x j 1 Whenα 1, F α G; x 0,,x j 1 and P α G; x 0,,x j 1 are reduced to F G; x 0,,x j 1 and P G; x 0,,x j 1, respectively From the definition, it is clear that the period of a polytopic--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 46 Let G be a finite group If there exists a polytopic--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--nacci sequenceable It is important to note that the direct product of polytopic--nacci sequenceable groups is not necessarily polytopic--nacci sequenceable Consider that the group C C 4 is defined by the presentation x, y x y 4 e, xy yx 45 The polytopic Fibonacci sequences of the group C C 4 for α are F ( C C 4 ; x, y ) x, y, xy,y 3,x,y,, F ( C C 4 ; y, x ) y, x, y 3,xy,y,x, 46 Since the elements e, xy, and xy 3 do not in either sequences, the group C C 4 is not polytopic--nacci sequenceable The group x has a polytopic Fibonacci sequence F x ; e, x e,x,e,x, 47 and hence is polytopic--nacci sequenceable The group y has a polytopic Fibonacci sequence F ( y ; e, y ) e, y, y,y 3,e,y, 48 and hence is polytopic--nacci sequenceable We will now address the periods of the polytopic--nacci sequences in specific classes of groups A group SD m is semidihedral group of order m if SD m a, b a m 1 b e, b 1 ab a 1 m 49 for every m 4 Note that the orders a and b are m 1 and, respectively

10 10 Discrete Dynamics in Nature and Society Theorem 47 The periods of the polytopic--nacci sequences in the group SD m for initial (seed) set, a, b, andα areasfollows: i P SD m; a, b h m, for 4 ii P SD m; a, b h m 1, for 5 Proof i If, we have the polytopic--nacci sequence for α : x 0 a, x 1 b, x a 3, x 3 a m 1 3 b, x 4 a 3, x 5 a m 1 3 m 1 3 b,, 410 x m a 3m 3, x m 1 a m 1 3 m 1 3 m 1 3 m 3 b, By mathematical induction, it is easy to prove that ( 3 m 3 1 mod m 1) ( ) ( ), m 1 3 m 1 3 ( ) ( m 1 3 m 3 0 mod m 1) 411 So we get x m a 3m 3 a, x m 1 a m 1 3 m 1 3 m 1 3 m 3 b b It is easy to see that h m m 3 h m 3 m Since the elements succeeding x h m, x h, depend on a and b for their values, the cycle begins again with the h m 1 m nd, that is, x h m x 0 and x h m 1 x 1Thus,theperiodofF SD m; a, b is h m If 3, we have the polytopic-3-nacci sequence for α : x 0 a, x 1 b, x a 3, x 3 a 4 m 1 3 b, x 4 a 3, x 5 a m 1 3 m 1 3 b, x 6 a 33,, x m a 3m 3, x m 1 a 3m m m 1 3 m 1 3 m 1 3 m 3 b, x m a 3m By mathematical induction, it is easy to prove that 3 m m mod m 1, 3 m mod m 1 So we get x m a 3m 3 a, x m 1 a 3m m m 1 3 m 1 3 m 1 3 m 3 b b, x m a 3m 3 1 a 3 Itiseasy to see that h 3 m m 3 h 3 m 3 m Since the elements succeeding x h 3 m,x h 3 m 1,x h 3 m depend on a, b, anda3 for their values, the cycle begins again with h 3 m nd, that is x h 3 m x 0,x h 3 m 1 x 1,andx h 3 m 1 x Thus,theperiod of F 3 SD m; a, b is h 3 m The proof for 4 is similar and is omitted

11 Discrete Dynamics in Nature and Society 11 ii If 5, we have the polytopic--nacci sequence for α : x 0 a, x 1 b, x a 3, x 3 a m 1 b, x 4 a 14, x 5 a u 1, x 6 a u,,x a u 4, x 4h a5, x 4h 1 a8 b, x 4h a7, x 4h 3 am 1 b, x 4h 4 a30, x 4h 5 au 1 8 λ 1, x 4h 8 λ,, x 4h 4 8 λ 4,, x i 4h a1 4 i, x i 4h 1 a8 i b, x i4h a3 4 i, 413 x i 4h 3 am 1 b, x i 4h 4 a14 16 i, x i 4h 5 au 1 8 i λ 1, x i 4h 6 au 8 i λ,,x i 4h au 4 8 i λ 4,, where λ 1,,λ 4 are natural numbers and u 1,,u 4 are even natural numbers So we need the smallest i N such that 8 i m 1 If we choose i m 4,weobtainx h a m 1 x 0, x h m 1 1 b x 1, x h m 1 a3 x, x h x h m 1 5 au 1 x 5, x h m 1 6 au x 6,, x h h m 1 SowegetP SD m; a, b h m 1 for 5 m 1 3 am 1 b x 3, x h m 1 4 a14 x 4, m 1 au 4 x since m h Theorem 48 The periods of the the polytopic--nacci sequences in the group SD m for initial (seed) sets b, a, andα areasfollows: i P SD m; b, a h m for 3, ii P SD m; b, a h m 1 for 4 Proof The proof is similar to the proof of Theorem 45 and is omitted Acnowledgment This project was supported by the Commission for the Scientific Research Projects of Kafas University The Project no 010-FEF-61 References 1 E Kiliç and D Tasci, On families of bipartite graphs associated with sums of generalized order- Fibonacci and Lucas numbers, Ars Combinatoria, vol 94, pp 13 3, 010 G Y Lee, -Lucas numbers and associated bipartite graphs, Linear Algebra and its Applications, vol 30, no 1-3, pp 51 61, K Lü and W Jun, -step Fibonacci sequence modulo m, Utilitas Mathematica, vol 71, pp , E F Miles, Generalized Fibonacci numbers and associated matrices, The American Mathematical Monthly, vol 67, pp , 1960

12 1 Discrete Dynamics in Nature and Society 5 M Mouline and M Rachidi, -generalized Fibonacci sequences and Marov chains, The Fibonacci Quarterly, vol 38, no 4, pp , D Kalman, Generalized Fibonacci numbers by matrix methods, The Fibonacci Quarterly, vol 0, no 1, pp 73 76, C M Campbell and P P Campbell, The Fibonacci length of certain centro-polyhedral groups, Applied Mathematics and Computing, vol 19, no 1-, pp 31 40, A Sadeghieh and H Doostie, Non-abelian sequenceable groups involving α-covers, Sciences, vol 0, no 3, pp 77 8, S W Knox, Fibonacci sequences in finite groups, The Fibonacci Quarterly, vol 30, no, pp , H Aydin and R Diici, General Fibonacci sequences in finite groups, The Fibonacci Quarterly, vol 36, no 3, pp 16 1, 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 7 35, Kluwer Academic publisher, Dodrecht, The Netherlands, O Deveci and E Karaduman, On the basic -nacci sequences in finite groups, Discrete Dynamics in Nature and Society, vol 34, no 4, pp 75 83, 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-, pp 53 58, D D Wall, Fibonacci series modulo m, The American Mathematical Monthly, vol 67, pp 55 53, H J Wilcox, Fibonacci sequences of period n in groups, The Fibonacci Quarterly, vol 4, no 4, pp , A S Fraenel and S T Klein, Robust universal complete codes for transmission and compression, Discrete Applied Mathematics, vol 64, no 1, pp 31 55, G R Kaluge, Penggunaan fibonacci dan Josephus problem dalam algoritma enripsi transposisi substitusi, Maalah IF 3058 Kriptografi-Sem II Tahun 010/ 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 81 85, B K Gandhi and M J Reddy, Triangular numbers in the generalized associated Pell sequence, Indian Pure and Applied Mathematics, vol 34, no 8, pp , 003

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