NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCASBALANCING NUMBERS


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1 International Journal of Pure and Applied Mathematics Volume 85 No , ISSN: (printed version); ISSN: (online version) url: doi: PAijpam.eu NEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCASBALANCING NUMBERS Prasanta Kumar Ray International Institute of Information Technology Gothapatna, PO: MALIPADA, Bhubaneswar, , INDIA Abstract: Balancing numbers n and balancers r are originally defined as the solution of the Diophantine equation (n 1) = (n + 1) + (n + )+ +(n+r). If n is a balancing number, then 8n +1 is a perfect square. Further, If n is a balancing number then the positive square root of 8n +1 is called a Lucasbalancing number. These numbers can be generated by the linear recurrences B n+1 = 6B n B n 1 and C n+1 = 6C n C n 1 where B n and C n are respectively denoted by the n th balancing number and n th Lucasbalancing number. In this study, we establish some new identities for the common factors of both balancing and Lucasbalancing numbers. AMS Subject Classification: 11B39, 11B83 Key Words: balancing numbers, Lucasbalancing numbers, recurrence relation 1. Introduction Behera and Panda [1] recently introduced a number sequence called balancing numbers defined in the following way: A positive integer n is called a balancing number with balancer r, if it is the solution of the Diophantine equation (n 1) = (n+1)+(n+)+...+(n+r). They also proved that the Received: January 11, 013 c 013 Academic Publications, Ltd. url:
2 488 P.K. Ray recurrence relation for balancing numbers is B n+1 = 6B n B n 1, n >, (1.1) where B n is the n th balancing number with B 1 = 1 and B = 6. It is well known that (see [1]), n is a balancing number if and only if n is a triangular number, that is 8n +1 is a perfect square. In [10], Lucasbalancing numbers are defined as follows: If n is a balancing number, C n = 8n +1 is called a Lucasbalancing number. The recurrence relation for Lucasbalancing numbers is same as that of balancing numbers, that is C n+1 = 6C n C n 1, n >, (1.) where C n is the n th Lucasbalancing number with C 1 = 3 and C = 17. Liptai [4], showed that the only balancing number in the sequence of Fibonacci numbers is 1. In [11] and [1], Ray obtain nice product formulas for both balancing and Lucasbalancing numbers. Panda and Ray [8], link balancing numbers with Pell and associated Pell numbers. They shown that balancing numbers are indeed the product of Pell and associated Pell numbers. Many interesting properties and important identities are available in the literature. Interested readers can follow [, 3, 5, 6, 7, 13, 14]. The closed form of both balancing and Lucasbalancing numbers are respectively given by and B n = λn 1 λn λ 1 λ (1.3) C n = λn 1 +λn (1.4) for n 1 with λ 1 = 3 + 8, λ = 3 8. These relations (1.3) and (1.4) are popularly known as Binet s formulas for balancing and Lucasbalancing numbers. In this paper, we obtain some new identities for the common factors of these numbers.. New Identities for the Common Factors of Balancing and LucasBalancing Numbers In this section, we present some new identities for the common factors of both balancing and Lucasbalancing numbers with the help of Binet s formula. It is clear that λ 1 +λ = 6, λ 1 λ = 8, λ 1 λ = 1. (.1)
3 NEW IDENTITIES FOR THE COMMON FACTORS OF Theorem.1. For n 1, the following identity is valid: Proof. By (.1), we obtain which finishes the proof. B 4n 6 = B n 1 C n+1. B n 1 C n+1 = λn 1 1 λ n 1 λ n+1 λ 1 λ = λ4n 1 λ4n λ 1 λ λ 1 λ λ 1 λ = B 4n 6 1 +λ n+1 Theorem.. For n 1, the following identity is valid: Proof. By (.1), we get B 4n+1 +1 = B n+1 C n. B n+1 C n = λn+1 1 λ n+1 λ n 1 +λ n λ 1 λ = λ4n+1 1 λ 4n+1 + (λ 1λ ) n λ 1 (λ 1 λ ) n λ λ 1 λ λ 1 λ = B 4n+1 +1 which is the end of the proof. Theorem.3. For n 1, the following identity is valid: Proof. By (.1), we have which is the end of the proof. B 4n+ +6 = B n+ C n. B n+ C n = λn+ 1 λ n+ λ n 1 +λ n λ 1 λ = λ4n+ 1 λ 4n+ +(λ 1 λ ) nλ 1 λ λ 1 λ λ 1 λ = B 4n+ +6
4 490 P.K. Ray By the same way, we have the following result. Theorem.4. For n 1, the following identity is valid: B 4n+3 1 = B n+1 C n+. The following lemma is already established in[8]. For the sake of simplicity we present the proof again. Lemma.5. For n 1, the following identity is valid: Proof. By (.1), we get which completes the proof. B n = B n C n. B n C n = λn 1 λn λ n 1 +λn λ 1 λ = λn 1 λn λ 1 λ = B n Lemma.6. For n 1, the following identity is valid: Proof. By (.1), we have which is the end of the proof. B 4n+1 1 = B n C n+1. B n C n+1 = λn 1 λn λ n+1 λ 1 λ 1 +λ n+1 = λ4n+1 1 λ 4n+1 (λ 1 λ ) nλ 1 λ λ 1 λ λ 1 λ = B 4n+1 1 By virtue of Lemma.5 and Lemma.6, we have the following result. Corollary.7. For n 1, we have B 4n+1 1 = B n C n C n+1.
5 NEW IDENTITIES FOR THE COMMON FACTORS OF Lemma.8. For n 1, the following identity is valid: C 4n+1 3 = 16B n B n+1. Proof. Since (λ 1 λ ) = 3, we get which ends the proof. 16B n B n+1 = 16 λn 1 λn λ 1 λ λ n+1 = λ4n+1 1 +λ 4n+1 = C 4n λ n+1 λ 1 λ (λ 1 λ ) nλ 1 +λ Since B n = B n C n, the following identity is valid for n 1: Theorem.9. C 4n+1 3 = C n C n+1. Theorem.10. For n 1, the following identity is valid: Proof. By (.1), we have which is the end of the proof. C 4n+1 +3 = C n C n+1. C n C n+1 = λn 1 +λn λ n+1 = λ4n+1 1 +λ 4n+1 = C 4n λ n+1 +(λ 1 λ ) nλ 1 +λ Lemma.11. For n 1, the following identity is valid: Proof. By using (.1), we have which completes the proof. B 4n+3 +1 = B n+ C n+1. B n+ C n+1 = λn+ 1 λ n+ λ n+1 λ 1 λ 1 +λ n+1 = λ4n+ 1 λ 4n+ +(λ 1 λ ) n+1 λ 1 λ = B 4n+3 +1
6 49 P.K. Ray Theorem.1. For n 1, we have B 4n+3 +1 = 4B n+1 C n+1 C n+1. Proof. SubstitutingB n+ = B n+1 C n+1 fromlemma.5intolemma.11, we obtain the desired result. Theorem.13. For n 1, the following identity is valid: Proof. By (.1), we have C 4n+3 +3 = C n+1 C n+. C n+1 C n+ = λn+1 1 +λ n+1 λ n+ which completes the proof. = λ4n+3 1 +λ 4n+3 = C 4n λ n+ +(λ 1 λ ) n+1λ 1 +λ Theorem.14. For n 1, the following identity is valid: C 4n+3 3 = 16B n+1 B n+. Proof. Since (λ 1 λ ) = 3, we obtain which ends the proof. 16B n+1 B n+ = λn+1 1 λ n+1 λ 1 λ ) = λ4n+3 1 +λ 4n+3 = C 4n+3 3 λ n+ 1 λ n+ λ 1 λ (λ 1 λ ) n+1λ 1 +λ The following corollary is an immediate consequence of Theorem.14. Corollary.15. For n 1, we have C 4n+3 3 = 3B n+1 C n+1 B n+1.
7 NEW IDENTITIES FOR THE COMMON FACTORS OF References [1] A. Behera and G.K. Panda, On the square roots of triangular numbers, The Fibonacci Quarterly, 37, No. (1999), [] A. Berczes, K. Liptai, I. Pink, On generalized balancing numbers, Fibonacci Quarterly, 48, No. (010), [3] R. Keskin, O. Karaatly, Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15, No. 1 (01). [4] K. Liptai, Fibonacci balancing numbers, The Fibonacci Quarterly, 4, No. 4 (004), [5] K. Liptai, Lucas balancing numbers, Acta Math.Univ. Ostrav, 14, No. 1 (006), [6] K. Liptai, F. Luca, A. Pinter, L. Szalay, Generalized balancing numbers, Indagationes Math. N. S., 0 (009), [7] P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37 (010), [8] G.K. Panda, P.K. Ray, Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 6, No. 1 (011), [9] G.K. Panda, P.K. Ray, Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 8 (005), [10] G.K. Panda, Some fascinating properties of balancing numbers, Proc. Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194 (009), [11] P.K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucasbalancing numbers, Bol. Soc. Paran. Mat., 30, No. (01), [1] P.K. Ray, Factorization of negatively subscripted balancing and Lucasbalancing numbers, Bol.Soc.Paran.Mat., 31, No. (013), [13] P.K. Ray, Curious congruences for balancing numbers, Int. J. Contemp. Sciences, 7, No. 18 (01),
8 494 P.K. Ray [14] P.K. Ray, Certain matrices associated with balancing and Lucasbalancing numbers, Matematika, 8, No. 1 (01), 15.
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