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[7] P.K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing numbers, Bol. Soc. Paran. Mat.30 (2), 2012, 49-56.
[8] P.K. Ray, Factorization of negatively subscripted balancing and Lucas-balancing numbers, Bol.Soc.Paran.Mat., 31 (2), 2013, 161-173.
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[10] G.K. Panda and 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(1), 2011, 41-72.
[11] P. K. Ray, Certain matrices associated with balancing and Lucas-balancing numbers, Matematika, 28 (1), 2012, 15-22.
[12] K. Liptai, Lucas balancing numbers, Acta Math.Univ. Ostrav, 14(1), 2006,43-47.
[13] K. Liptai, F. Luca, A. Pinter and L. Szalay, Generalized balancing numbers, Indagationes Math. N. S., 20, 2009, 87-100.
[14] R. Keskin and O. Karaatly, Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15(1), 2012.
[15] P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37, 2010, 125-138.
[16] G.K. Panda and P.K. Ray, Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 2005(8), 2005, 1189-1200.
[17] P.K. Ray, Curious congruences for balancing numbers, Int.J.Contemp.Sciences, 7 (18), 2012, 881-889.
[18] P.K Ray, New identitities for the common factors of balancing and Lucas-balancing numbers, International Journal of Pure and Applied Mathematics, 85(3), 2013, 487-494.
[19] P.K. Ray, Some congruences for balancing and Lucas-balancing numbers and their applications, Integers, 14, 2014, #A8.
[20] P.K. Ray, Balancing sequences of matrices with application to algebra of balancing numbers, Notes on Number Theory and Discrete Mathematics, 20(1), 2014, 49-58.[20] P.K. Ray, On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, 3(1), 2014, 1-6.
[2] A.P. Stakhov, The golden matrices and a new kind of cryptography, Chaos, Soltions and Fractals32, 2007, 1138–1146.
[3] A. Luma and B. Raufi, Relationship between Fibonacci and Lucas Sequences and their Application in Symmetric Cryptosystems, Latest Trends on Circuits, Systems and Signals, 4th International Conference on Circuits, Systems and Signals, July 22-25, 2010, 146-150
[4] A. Behera and G.K. Panda, On the square roots of triangular numbers, The Fibonacci Quarterly, 37(2), 1999, 98-105.
[5] G.K. Panda, Some fascinating properties of balancing numbers, Proc. Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, 2009, 185-189.
[6] K. Liptai, Fibonacci balancing numbers, The Fibonacci Quarterly, 42(4), 2004, 330-340.
[7] P.K. Ray, Application of Chybeshev polynomials in factorization of balancing and Lucas-balancing numbers, Bol. Soc. Paran. Mat.30 (2), 2012, 49-56.
[8] P.K. Ray, Factorization of negatively subscripted balancing and Lucas-balancing numbers, Bol.Soc.Paran.Mat., 31 (2), 2013, 161-173.
[9] P.K. Ray, Factorization of negatively subscripted balancing and Lucas-balancing numbers, Bol.Soc.Paran.Mat., 31 (2), 2013, 161-173.
[10] G.K. Panda and 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(1), 2011, 41-72.
[11] P. K. Ray, Certain matrices associated with balancing and Lucas-balancing numbers, Matematika, 28 (1), 2012, 15-22.
[12] K. Liptai, Lucas balancing numbers, Acta Math.Univ. Ostrav, 14(1), 2006,43-47.
[13] K. Liptai, F. Luca, A. Pinter and L. Szalay, Generalized balancing numbers, Indagationes Math. N. S., 20, 2009, 87-100.
[14] R. Keskin and O. Karaatly, Some new properties of balancing numbers and square triangular numbers, Journal of Integer Sequences, 15(1), 2012.
[15] P. Olajos, Properties of balancing, cobalancing and generalized balancing numbers, Annales Mathematicae et Informaticae, 37, 2010, 125-138.
[16] G.K. Panda and P.K. Ray, Cobalancing numbers and cobalancers, International Journal of Mathematics and Mathematical Sciences, 2005(8), 2005, 1189-1200.
[17] P.K. Ray, Curious congruences for balancing numbers, Int.J.Contemp.Sciences, 7 (18), 2012, 881-889.
[18] P.K Ray, New identitities for the common factors of balancing and Lucas-balancing numbers, International Journal of Pure and Applied Mathematics, 85(3), 2013, 487-494.
[19] P.K. Ray, Some congruences for balancing and Lucas-balancing numbers and their applications, Integers, 14, 2014, #A8.
[20] P.K. Ray, Balancing sequences of matrices with application to algebra of balancing numbers, Notes on Number Theory and Discrete Mathematics, 20(1), 2014, 49-58.[20] P.K. Ray, On the properties of Lucas-balancing numbers by matrix method, Sigmae, Alfenas, 3(1), 2014, 1-6.
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Copyright
© Computer Engineering and Applications Journal, 2016
Affiliations
Sujata Swain
Department of Computer Science, DAV Unit-8, Bhubaneswar
Chidananda Pratihary
Department of Mathematics, National Institute of Technology, Rourkela
Prasanta Kumar Ray
Department of Mathematics, VSS University of Technology, Burla
Balancing and Lucas-Balancing Numbers and their Application to Cryptography
Abstract
It is well known that, a recursive relation for the sequence Â is an equation that relates Â to certain of its preceding terms . Initial conditions for the sequence Â are explicitly given values for a finite number of the terms of the sequence. The recurrence relation is useful in certain counting problems like Fibonacci numbers, Lucas numbers, balancing numbers, Lucas-balancing numbers etc. In this study, we use the recurrence relations for both balancing and Lucas-balancing numbers and examine their application to cryptography.