Generalized balancing numbers

Properties of balancing, cobalancing and generalized balancing numbers∗

A positive integer n is called a balancing number if 1 + 2 + · · · + (n − 1) = (n + 1) + (n + 2) + · · · + (n + r) for some positive integer r. Several authors investigated balancing numbers and their various generalizations. The goal of this paper is to survey some interesting properties and results on balancing, cobalancing and all types of generalized balancing numbers.

On the Properties of Balancing and Lucas-Balancing $p$-Numbers

The main goal of this paper is to develop a new generalization of balancing and Lucas-balancing sequences namely balancing and Lucas-balancing $p$-numbers and derive several identities related to them. Some combinatorial forms of these numbers are also presented.

Application of Balancing Numbers in Effectively Solving Generalized Pell’s Equation

Solving a generalized Pell’s equation of the form , basically involves two steps. The first step is to find out the primitive solutions for the same, and the second step is to solve related Pell’s equation ; and combine the two solutions. Therefore it is evident that the number of independent primitive solutions determines the number of independent solution sets. In our work, we have tried to r...

On generalized fuzzy numbers

This paper first improves Chen and Hsieh’s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the λ-cutting sets or generalized λ-cutting sets, so that the generalized fuzzy numbers are integrat...

Addition two generalized fuzzy numbers