On Some Identities of k-fibonacci Sequences Modulo Ring Z 6 and Z 10

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 9, HIKARI Ltd, On Some Identities of k-fibonacci Sequences Modulo Ring Z 6 and Z 10 Tri Wahyuni, Sri Gemawati and Syamsudhuha Department of Mathematics University of Riau, Pekanbaru 28293, Indonesia Corresponding author Copyright c 2018 Tri Wahyuni, Sri Gemawati and Syamsudhuha. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This study discusses the k-fibonacci sequence modulo ring Z 6 and Z 10. Some of k-fibonacci sequences modulo ring Z 6 and Z 10 have correlation with k-fibonacci sequence modulo Z 2, Z 3 and Z 5. Keywords: k-fibonacci Sequence, Modulo Arithmetic, Period-Length, Identity of k-fibonacci Modulo Ring Z 6 and Z 10 1 Introduction Many papers and research works have been dedicated to Fibonacci numbers and their generalizations. Varnadore [11] examined the relationship of Fibonacci numbers and Pascal s triangle. Vince [12], Andreassian [1], Ehrlich [5] and Chang [4] discussed some properties of Fibonacci modulo n. Furthermore, Falcon and Plaza, [6],[7], [8], Kose [2] and Catarino [3] conducted some researches on k-fibonacci numbers. Afterwards, Falcon and Plaza [9] proved some properties of the sequences of k-fibonacci modulo m, in which in this paper, it is noticed by Fk,n m. In 2013, Leyandekker and Shannon [10] analyzed the structure of Fibonacci numbers at modulo ring Z 5. Following the idea of Leyandekker and Shannon, in this study, it is expanded the sequences of k-fibonacci modulo ring Z 6 and Z 10.

2 442 Tri Wahyuni, Sri Gemawati and Syamsudhuha Since 2 and 3 are the factors of 6, the k-fibonacci sequence modulo ring Z 2 and Z 3 have correlation with Z 6, and the same case applies to k-fibonacci sequence modulo ring Z 10. Falcon and Plaza define the k-fibonacci sequence [6] as follows: Definition 1.1 For any integer number k 1, the k th Fibonacci sequence, say {F k,n } n N is defined recurrently by F k,0 = 0, F k,1 = 1, and F k,n+1 = kf k,n + F k,n 1 for n 1. (1) For k = 1, 2, 3, 4, k-fibonacci sequence (1.1) are as follows: {F 1,n } n N = {0, 1, 1, 2, 3, 5, 8, 13, } {F 2,n } n N = {0, 1, 2, 5, 12, 29, 70, } {F 3,n } n N = {0, 1, 3, 10, 33, 109, 360, } {F 4,n } n N = {0, 1, 4, 17, 72, 305, 1292, } Furthermore, Falcon and Plaza proved some of the following properties k-fibonacci sequences (see [6],[7], [8] for details): (i) First combinatorial formula for general term F k,n = 1 2 n 1 n 1 2 i=1 ( ) n k n 1 2i (k 2 + 4) i. 2i + 1 (ii) Second combinatorial formula for general term F k,n = (iii) Catalan s identity (iv) Cassini s identity n 1 2 i=1 ( n 1 i i ) k n 1 2i, untukn 2. F k,n r F k,n+r F 2 k,n = ( 1) n+1 r F 2 k,r. F k,n 1 F k,n+1 F 2 k,n = ( 1) n. (v) Sum of the first n terms k-fibonacci sequence n F k,i = 1 k (F k,n+1 + F k,n 1). i=1

3 On some identities of k-fibonacci sequences modulo ring Z 6 and Z (vi) Sum of the first n even terms for k-fibonacci sequence n F k,2i = 1 k (F k,2n+1 1). (vii) Sum of the first n odd terms k-fibonacci sequence n F k,2i+1 = 1 k (F k,2n+2). i=1 i=0 (viii) d Ocagne s identity F k,m F k,n+1 F k,m+1 F k,n = ( 1) n F k,m n jika m > n. (ix) Generating function F k (x) = x 1 kx x 2. 2 k-fibonacci Sequences Modulo Ring Z 2, Z 3, Z 5, Z 6 and Z 10 In this section, some table of k-fibonacci sequences modulo ring Z 2, Z 3, Z 5, Z 6 and Z 10 are displayed. Those were used in this study. Table 1: The First Three Period of k-fibonacci Modulo 2 n F1,n F(2,n) Table 1 shows the first three periods of k-fibonacci sequences modulo ring Z 2. 1-Fibonacci sequences modulo Z 2 is the repetiton of {0,1,1} and 2-Fibonacci sequences modulo ring Z 2 is the repetiton of {0,1}. Table 2: The First Two Period k-fibonacci Modulo 3 n F(1,n) F(2,n)

4 444 Tri Wahyuni, Sri Gemawati and Syamsudhuha Table 2 shows the first two period of the k-fibonacci sequences modulo ring Z 3. 1-Fibonacci sequence modulo ring Z 3 is the repetition of { 0, 1, 1, 2, 0, 2, 2, 1}, and 2-Fibonacci sequence modulo ring Z 3 is the repetition of {0, 1, 2, 2, 0, 2, 1, 1}, so, the period length of the 1-Fibonacci sequence modulo ring Z 3 is 8, same as 2-Fibonacci modulo ring Z 3. Table 3: The First Period k-fibonacci Sequence Modulo 5 n F1,n n F1,n n F2,n F3,n F4,n n F4,n Table 3 displays k-fibonacci sequence modulo ring Z 5. Period-length for k = 1 is 20, same as k = 4 and period-length for k = 2 is 12 same as k = 3. Table 4: The First Period k-fibonacci Sequence Modulo 6 n F(1,n) n F(1,n) n F(2,n) F(3,n) F 6 (4,n) n F(5,n) n F(5,n) Table 4 shows the period-length of k-fibonacci sequence modulo ring Z 6 for k = 1 same as k = 5 and period-length for k = 2 same as k = 4. Table 5 shows the period-length of k-fibonacci sequence modulo ring Z 10 for k = 1 and k = 2. Period-length for k = 1 is 60 and for k = 2 is 12.

5 On some identities of k-fibonacci sequences modulo ring Z 6 and Z Table 5: The First Period k-fibonacci Sequence Modulo 10 for k = 1 and 2 n F(1,n) n F(1,n) n F(1,n) n F(1,n) n F(2,n) The Identities of k-fibonacci sequences modulo Ring Z 6 and Z 10 Some of k-fibonacci sequence modulo ring Z 6 have correlation with k-fibonacci sequence modulo ring Z 2 and Z 3, while k-fibonacci sequence modulo ring Z 10 have correlation with k-fibonacci sequence modulo ring Z 2 and Z 5. As a result, some of identities for k-fibonacci sequence modulo ring Z 6 and Z 10 are derived. F(1,n) 6 has correlation with F (1,n) 2 and F (1,n) 3. Table 6 shows the correlation among 1-Fibonacci sequence modulo ring Z 2, Z 3 and Z 6. Table 6: The Congruence 3F 2 (1,n) + 4F 3 (1,n) F 6 (1,n) n F(1,n) 2 F(1,n) 3 3F(1,n) 2 + 4F (1,n) 3 F(1,n) Theorem 3.1 For each natural number n, 3F 2 (1,n) + 4F 3 (1,n) F 6 (1,n). Proof. Theorem 3.1 is true for n=1, and will be showed that it is true for n = p + 1 by using Definition 1.1 F k,n+1 = kf k,n + F k,n 1, untuk n 1,

6 446 Tri Wahyuni, Sri Gemawati and Syamsudhuha 3F 2 1, p+1 + 4F 3 1, n+1 3(F 2 (1, p) + F 2 (1, p 1)) + 4(F 3 (1, p) + F 3 (1, p 1)) 3F 2 (1, p) + 3F 2 (1, p 1) + 4F 3 (1, p) + 4F 3 (1, p 1) 3F 2 (1, p) + 4F 3 (1, p) + 3F 2 (1, p 1) + 4F 3 (1, p 1) F 6 (1,p) + F 6 (1,p 1) F 6 (1,p+1) So that, 3F 2 (1,n) + 4F 3 (1,n) F 6 (1,n) is proved. F(2,n) 6 has correlation with F (2,n) 2 and F (2,n) 3. Table 7 shows the correlation among 2-Fibonacci sequence modulo ring Z 2, Z 3 and Z 6, so Theorem 3.2 is obtained. Table 7: The Congruence 3F 2 (2,n) + 4F 3 (2,n) F 6 (2,n) n F(2,n) 2 F(2,n) 3 3F(2,n) 2 + 4F (2,n) 3 F(2,n) Theorem 3.2 For each natural number n, 3F 2 (2,n) + 4F 3 (2,n) F 6 (2,n). Proof. The proof of Theorem 3.2 is similar to the proof of Theorem 3.1. F(1,n) 10 has correlation with F (1,n) 2 and F (1,n) 5. Table 8 displays the correlation among 1-Fibonacci sequence modulo ring Z 2, Z 5 and Z 10, then the identity of 1-Fibonacci sequence modulo ring Z 10 in Theorem 3.3 is obtained. Theorem 3.3 For each natural number n, 5F 2 (1,n) + 6F 5 (1,n) F 10 (1,n). Proof. The proof of Theorem 3.3 is similar to the proof of Theorem 3.1. F(2,n) 10 has correlation with F (2,n) 2 and F (2,n) 5. Table 9 displays the correlation among 2-Fibonacci sequence modulo ring Z 2, Z 5 and Z 10, then the identity of 2-Fibonacci sequence modulo ring Z 10 in Theorem 3.3 is obtained. Theorem 3.4 n, 5F 2 (2,n) + 6F 5 (2,n) F 10 (2,n). Proof. The proof of Theorem 3.4 is similar to the proof of Theorem 3.1.

7 On some identities of k-fibonacci sequences modulo ring Z 6 and Z Table 8: The Congruence 5F 2 (1,n) + 6F 5 (1,n) F 10 (1,n) n F(1,n) 2 F(1,n) 5 5F(1,n) 2 + 6F (1,n) 5 F(1,n) Table 9: The Congruence 5F 2 (2,n) + 6F 5 (2,n) F 10 (2,n) n F(2,n) 2 F(2,n) 5 5F(2,n) 2 + 6F (2,n) 5 F(2,n) Conclusion To conclude, some k-fibonacci sequences modulo 6 have correlation with k- Fibonacci sequences modulo 2 and 3, that are 3F(1,n) 2 + 4F (1,n) 3 F(1,n) 6 and 3F(2,n) 2 + 4F (2,n) 3 F(2,n) 6. Meanwhile, some k-fibonacci sequences modulo 10 have correlation with k-fibonacci sequences modulo 2 and 5, that are 5F(1,n) 2 + 6F(1,n) 5 F (1,n) 10 and 5F (2,n) 2 + 6F (2,n) 5 F (2,n) 10. References [1] A. Andreassian, Fibonacci Sequences Modulo m, Fibonacci Quarterly, 12 (1974),

8 448 Tri Wahyuni, Sri Gemawati and Syamsudhuha [2] C. Bolat and H. Kose, On The Properties of k-fibonacci Numbers, International Journal of Contemporary Mathematical Science, 5 (2010), [3] P. Catarino, On Some Identities for k-fibonacci Sequence, International Journal of Contemporary Mathematical Science, 9 (2014), [4] D. K. Chang, Higher-Order Fibonacci Sequences Modulo M, Fibonacci Quarterly, 7 (1984), [5] A. Ehrlich, On The Periods of The Fibonacci Sequence Modulo n, Fibonacci Quarterly, 27 (1989), [6] S. Falcon and A. Plaza, On The Fibonacci k-numbers, Chaos, Solitons and Fractals, 32 (2007), [7] S. Falcon and A. Plaza, The k-fibonacci Sequence and The Pascal 2- Triangle, Chaos, Solitons and Fractals, 33 (2007), [8] S. Falcon and A. Plaza, On k-fibonacci Sequence and Polynimials and Their Derivates, Chaos, Solitons and Fractals, 39 (2009), [9] S. Falcon and A. Plaza, k-fibonacci Sequences Modulo m, Chaos, Solitons and Fractals, 41 (2009), [10] J. V. Leyandekker, A.G. Shannon, The Structure of The Fibonacci Numbers in The Modular Ring Z5, Note on Number Theory and Discret Mathematics, 1 (2013), [11] J. Varnadore, Pascal s Triangle and Fibonacci Numbers, The Mathematics Theacher, 84 (1991), no. 4, [12] A. Vince, The Fibonacci Sequence Modulo n, The Fibonacci Quartely, 16 (1978), Received: February 23, 2018; Published: March 29, 2018