Tridiagonal matrices related to subsequences of balancing and Lucas-balancing numbers

Prasanta K Ray and Gopal K Panda
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 3, Pages 56–63
Full paper (PDF, 176 Kb)

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Authors and affiliations

Prasanta K Ray
Department of Mathematics, Veer Surendra Sai University of Technology
Odisha, Burla-768018, India

Gopal K Panda
Department of Mathematics, National Institute of Technology Rourkela
Rourkela-769008, India

Abstract

It is well known that balancing and Lucas-balancing numbers are expressed as determinants of suitable tridiagonal matrices. The aim of this paper is to express certain subsequences of balancing and Lucas-balancing numbers in terms of determinants of tridiagonal matrices. Using these tridiagonal matrices, a factorization of the balancing numbers is also derived.

Keywords

  • Balancing numbers
  • Balancers
  • Lucas-balancing numbers
  • Tridiagonal matrices

AMS Classification

  • 11B39
  • 11B83

References

  1. Behera, A., & Panda, G.K. (1999) On the square roots of triangular numbers, The Fibonacci Quarterly, 37(2), 98–105.
  2. Belbachair, H., & Szalay, L. (2014) Balancing in direction (1,−1) in Pascal’s triangle, Armenian Journal of Mathematics, 6(1), 32–40.
  3. Berczes, A., Liptai, K., & Pink, I. (2010) On generalized balancing numbers, Fibonacci Quarterly, 48(2), 121–128.
  4. Keskin R., & Karaatly, O. (2012) Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15(1).
  5. Liptai, K., Luca, F., Pinter, A., & Szalay, L. (2009) Generalized balancing numbers, Indagationes Math. N. S., 20, 87–100.
  6. Olajos, P. (2010) Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37, 125–138.
  7. 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.
  8. Panda, G. K. (2009) Some fascinating properties of balancing numbers, Proc. Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, 185–189.
  9. Ray, P. K. (2012) Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing numbers, Boletim da Sociedade Paranaense de Matematica, 30(2), 49–56.
  10. Ray, P. K. (2013) Factorization of negatively subscripted balancing and Lucas-balancing numbers, Boletim da Sociedade Paranaense de Matematica, Vol.31 (2), 161–173.
  11. Ray, P. K. (2012) Certain matrices associated with balancing and Lucas-balancing numbers, Matematika, 28(1), 15–22.
  12. Ray, P. K. (2013) New identities for the common factors of balancing and Lucas-balancing numbers, International Journal of Pure and Applied Mathematics, 85, 487–494.
  13. Ray, P. K. (2014) Some congruences for balancing and Lucas-balancing numbers and their applications, Integers, 14, #A8.
  14. Ray, P. K. (2014) Balancing sequences of matrices with application to algebra of balancing numbers. Notes on Number Theory and Discrete Mathematics, 20(1), 49–58.
  15. Ray, P. K. (2014) On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, 3(1), 1–6.
  16. Ray, P. K. (2014) Generalization of Cassini formula for balancing and Lucas-balancing numbers, Matematychni Studii., 42(1), 9–14.
  17. Ray, P. K. (2015) Balancing and Lucas-balancing sums by matrix methods, Mathematical Reports, 17(67), 2.

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Cite this paper

Ray, P. K., & Panda, G. K. (2015). Tridiagonal matrices related to subsequences of balancing and Lucas-balancing numbers. Notes on Number Theory and Discrete Mathematics, 21(3), 56-63.

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