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
Full paper (PDF, 158 Kb)

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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} = 6a_n − a_{n − 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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