Prasanta K Ray, Sunima Patel and Manoj K Mandal

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 4, Pages 41–48

**Full paper (PDF, 173 Kb)**

## Details

### Authors and affiliations

Prasanta K Ray

*Veer Surendra Sai University of Technology
Odisha, Burla, India*

Sunima Patel

*National Institute of Technology
Rourkela, India*

Manoj K Mandal

*National Institute of Technology
Rourkela, India*

### Abstract

It is well-known that the balancing numbers are the square roots of the triangular numbers and are the solutions of the Diophantine equation 1 + 2 + … + (*n* − 1) = (*n* + 1) + (*n* + 2) + … + (*n* + *r*), where *r* is the balancer corresponding to the balancing number *n*. Thus if *n* is a balancing number, then 8*n*^{2} + 1 is a perfect square and its positive square root is called a Lucas-balancing number. The goal of this paper is to establish some new identities of these numbers.

### Keywords

- Generating function
- Balancing
- Congruence

### AMS Classification

- 11B39
- 11B83

### References

- Alvarado, S., Dujella, A., & Luca, F. (2012) On a conjecture regarding balancing with powers of Fibonacci numbers, Integers, 12, 1127–1158.
- Behera, A., & Panda, G. K. (1999) On the square roots of triangular numbers, The Fibonacci Quarterly, 37(2), 98–105.
- Belbachair, H., & Szalay, L. (2014) Balancing in direction (1; –1) in Pascal’s triangle, Armenian Journal of mathematics, 6(1), 32–40.
- B´erczes, A., Liptai, K., & Pink, I. (2010) On generalized balancing sequences, The Fibonacci Quarterly, 48(2), 121–128.
- Irmak, N. (2013) Balancing with balancing powers, Miskolc Mathematical Notes, 14(3), 951–957.
- Keskin, R., & Karaatly, O. (2012) Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15(1), Article 12.1.4.
- Kovacs, T., Liptai, K., & Olajos, P. (2010) On (a; b)-type balancing numbers, Publicationes Mathematicae Debrecen, 77(3–4), 485–498.
- Olajos, P. (2010) Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37, 125–138.
- Panda, G. K., & Ray, P. K. (2011) Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 6(1), 41–72.
- Panda, G. K. (2009) Some fascinating properties of balancing numbers, Proceeding of the Eleventh International Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, 185–189.
- Panda, G. K., & Rout, S. S. (2013) Gap balancing numbers, The Fibonacci Quarterly, 51(3), 239–248.
- Panda, G. K., & Rout, S. S. (2014) Periodicity of Balancing numbers, Acta Math. Hungar., 143(2), 274–286.
- Ray, P. K. (2012) Certain matrices associated with balancing and Lucas-balancing numbers, MATEMATIKA, 28(1), 15–22.
- Ray, P. K. (2012) Curious congruences for balancing numbers, International Journal of Contemporary Mathematical Sciences, 7 (18), 881–889.
- Ray, P. K. (2014) Some congruences for balancing and Lucas-balancing numbers and their applications, Integers, 14, #A8.
- Ray, P. K. (2015) Balancing and Lucas-balancing sums by matrix methods, Mathematical Reports, 17(2), 225–233.

## Related papers

## Cite this paper

Ray, P.K., Patel, S. & Mandal, M. K. (2016). Identities for balancing numbers using generating function and some new congruence relations. *Notes on Number Theory and Discrete Mathematics*, 22(4), 41-48.