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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## Details

### Authors and affiliations

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 (*a*_{n + 1} = *ka _{n}*;

*n*= 0, 1, 2, …) and balancing (

*a*

_{n + 1}= 6

*a*

_{n}−

*a*

_{n − 1},

*a*

_{0}= 0,

*a*

_{1}= 1) admits an unique solution. In fact, the sequence is 1, λ

_{1}, λ

_{1}

^{2}, …, λ

_{1}

*, … where λ*

^{n}_{1}= 3 + √8 satisfies the balancing equation λ

^{2}− 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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## Related papers

## Cite this paper

APARay, 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.

ChicagoRay, 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.

MLARay, 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.