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

Prasanta Ray, Sunima Patel, Manoj Mandal
2016 Notes on Number Theory and Discrete Mathematics Print   unpublished
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 8n 2 + 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.