Identities for balancing numbers using generating function and some new congruence relations

Prasanta K Ray, Sunima Patel and Manoj K Mandal
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 4, Pages 41–48
Full paper (PDF, 173 Kb)

Details

Authors and affiliations

Prasanta K Ray
Veer Surendra Sai University of Technology
Odisha, Burla, India

Sunima Patel
National Institute of Technology
Rourkela, India

Manoj K Mandal
National Institute of Technology
Rourkela, India

Abstract

It is well-known that the balancing numbers are the square roots of the triangular numbers and are the solutions of the Diophantine equation 1 + 2 + … + (n − 1) = (n + 1) + (n + 2) + … + (n + r), where r is the balancer corresponding to the balancing number n. Thus if n is a balancing number, then 8n² + 1 is a perfect square and its positive square root is called a Lucas-balancing number. The goal of this paper is to establish some new identities of these numbers.

Keywords

• Generating function
• Balancing
• Congruence

AMS Classification

• 11B39
• 11B83

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