Some results on balancing, cobalancing,\( (a,b)\)-type balancing, and \((a,b)\)-type cobalancing numbers (Q2855597)

A positive integer \(n\) is said to be balancing (resp. cobalancing) if there exists a positive integer \(r\), the balancer (resp. cobalancer) of \(n\), such that \(1+2+\cdots+(n- 1)= (n+ 1)+ (n+ 2)+\cdots+(n+ r)\) (resp. \(1+2+\cdots+n= (n+ 1)+(n+ 2)+\cdots+ (n+ r)\)).NEWLINENEWLINE In the paper under review the authors prove new formulae for balancing and cobalancing numbers and for their \((a,b)\)-type generalizations. For coprime integers \(a\), \(b\) with \(a>0\) and \(b\geq 0\), \(an+b\) is sad to be \((a,b)\)-type balancing if there exists a positive integer \(r\) such that NEWLINE\[NEWLINE(a+b)+\cdots+(a(n- 1)+b)= (a(n+ 1)+b)+\cdots+ (a(n+ r)+b).NEWLINE\]NEWLINE For related results and references, see \textit{T. Kovács}, \textit{K. Liptai} and \textit{P. Olajos} [Publ. Math. 77, No. 3--4, 485--498 (2010; Zbl 1240.11053)].