Balances in the Set of Arithmetic Progressions

Abstract

This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included.

Keywords

• balancing numbers
• sequence balancing numbers
• (<i>a</i>,<i>b</i>)-type balancing numbers
• Pell equation