Almost balancing numbers (Q2798872)

In [Fibonacci Q. 37, No. 2, 98--105 (1999; Zbl 0962.11014)], \textit{A. Behera} and the first author defined balancing numbers \(n\) and balancers \(r\) as solutions of the Diophantine equation \(1 + 2 + 3 + \cdots + (n - 1) = (n + 1) + (n + 2) + \cdots + (n + r)\). By slightly modifying the above definition, the first author and \textit{P. K. Ray} [Int. J. Math. Math. Sci. 2005, No. 8, 1189--1200 (2005; Zbl 1085.11017)] defined cobalancing numbers \(n\) as solutions of the Diophantine equation \(1 + 2 + \cdots + n = (n + 1) + (n + 2) + \cdots + (n + r)\), calling \(r\) the cobalancer corresponding to \(n\). Behera and Panda [loc. cit.] proved that the square of any balancing number is a triangular number, while later the first author and Ray established the connection between triangular numbers of the form \(n^2 + n\) with cobalancing numbers. The concept of balancing numbers has been extended and generalized by many authors.NEWLINENEWLINEIn this paper with another mild modification of the defining equation for balancing numbers, the authors introduce the concept of almost balancing numbers.NEWLINENEWLINEThe authors call a natural number \(n\) an almost balancing number if NEWLINE\[NEWLINE |\left\{(n + 1) + (n + 2) + \cdots + (n + r)\right\} - \left\{1 + 2 + \cdots + (n - 1)\right\}| = 1,NEWLINE\]NEWLINE for some natural number \(r\) which we call the almost balancer corresponding to \(n\).NEWLINENEWLINEThis definition suggests that we can classify the almost balancing numbers in two classes. If NEWLINE\[NEWLINE \left\{(n + 1) + (n + 2) + \cdots + (n + r)\right\} - \left\{1 + 2 + \cdots + (n - 1)\right\} = 1, NEWLINE\]NEWLINE we call \(n\) an almost balancing number of \textit{first kind} while if NEWLINE\[NEWLINE \left\{(n + 1) + (n + 2) + \cdots + (n + r)\right\} - \left\{1 + 2 + \cdots + (n - 1)\right\} = -1, NEWLINE\]NEWLINE we call \(n\) an almost balancing number of \textit{second kind}. In former case, we call \(r\) an almost balancer of first kind, while in latter case we call r the almost balancer of second kind.NEWLINENEWLINEFor the sake of simplicity, they call almost balancing numbers of first kind as \(A_1\)-balancing numbers and almost balancers of first kind as \(A_1\)-balancers. Similarly, they call almost balancing numbers of second kind as \(A_2\)-balancing numbers and almost balancers of second kind as \(A_2\)-balancers.NEWLINENEWLINEOne of the most interesting results from this paper is the following:NEWLINENEWLINETheorem. If \(x\) is a balancing number then \(\alpha(x) = -5x + 2\sqrt{8x^2 + 1}\) and \(\beta(x) = -x + \sqrt{8x^2 + 1}\) are \(A_2\)-balancing numbers.NEWLINENEWLINEFor other interesting details see the original paper.