Balancing sequences of matrices with application to algebra of balancing numbers

Prasanta Kumar Ray
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 1, Pages 49—58
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Prasanta Kumar Ray
International Institute of Information Technology
Bhubaneswar, India

Abstract

It is well known that, the problem of finding a sequence of real numbers an, n = 0, 1, 2, …, which is both geometric (an + 1 = kan; n = 0, 1, 2, …) and balancing (an + 1 = 6anan − 1, a0 = 0, a1 = 1) admits an unique solution. In fact, the sequence is 1, λ1, λ12, …, λ1n, … where λ1 = 3 + √8 satisfies the balancing equation λ2 − 6λ + 1. In this paper, we pose an equivalent problem for a sequence of real, nonsingular matrices of order two and show that, this problem admits an infinity of solutions, that is there exist infinitely many such sequences.

Keywords

  • Balancing numbers
  • Lucas-balancing numbers
  • Balancing matrix
  • Lucas-balancing matrix

AMS Classification

  • 11B39
  • 11B83

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Cite this paper

APA

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.

Chicago

Ray, Prasanta Kumar. “Balancing Sequences of Matrices with Application to Algebra of Balancing Numbers.” Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 49-58.

MLA

Ray, Prasanta Kumar. “Balancing Sequences of Matrices with Application to Algebra of Balancing Numbers.” Notes on Number Theory and Discrete Mathematics 20.1 (2014): 49-58. Print.

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