Classes of gap balancing numbers (Q1981373)

In this work, the authors consider the class of gap balancing numbers. \textit{G. K. Panda} and \textit{S. S. Rout} [Fibonacci Q. 51, No. 3, 239--248 (2013; Zbl 1350.11038)] defined \(k\)-gap balancing numbers using the parity of the integer \(k\geq 0.\) When \(k\) is odd, a positive integer \(g_{k}\) is a \(k\)-gap balancing number if \[ 1+2+\cdots+(g_{k}-\frac{k+1}{2})=(g_{k}+\frac{k+1}{2})+(g_{k}+\frac{k+3}{2} )+\cdots+(g_{k}+r_{k}) \] for some integer \(r_{k}\geq 1\). When \(k\) is even, a positive integer \( g_{k}=2n+1\) is a \(k\)-gap balancing number if \[ 1+2+\cdots+(n-\frac{k}{2})=(n+\frac{k}{2}+1)+(n+\frac{k}{2}+^{\prime })+\cdots+(n+r_{k}) \] for some integer \(r_{k}\geq 0\). Then \(g_{k}\) is the median of the deleted gap when \(k\) is odd, and \(g_{k}\) is the sum of the two surviving numbers forming the edge of the gap when \(k\) is even. Let \(k\geq 0\) be an integer. A positive integer \(n\) is called an upper \( k\)-gap balancing number if \[1+2+\cdots+(n-k)=(n+1)+(n+2)+\cdots+(n+r) \] for some integer \(r\geq 0\). Here \(r\) is called the upper \(k\)-gap balancer corresponding to the upper \(k\)-gap balancing number \(n.\) It is clear that \(n\) is an upper \(k\)-gap balancing number if and only if \[ T(n-k)+T(n)=T(n+r), \] where \(T(n)\) is the \(n^{th}\) triangular number. If \(n\) is an upper \(k\)-gap balancing number, then from above equation \[ r=\frac{-2n-2+\sqrt{8n^{2}+8(1-k)n+(2k-1)^{2}}}{2} \] and \[ n=\frac{2r+2k-1+\sqrt{8r^{2}+8kr+1}}{2}. \] So \(n\) is an upper \(k\)-gap balancing number if and only if \( 8n^{2}+8(1-k)n+(2k-1)^{2}\) is a perfect square and hence \[ C=\sqrt{8n^{2}+8(1-k)n+(2k-1)^{2}} \] is an integer called upper \(k\)-gap Lucas-balancing number corresponding to \(n \) and \(r\) is an upper \(k\)-gap balancer if and only if \(8r^{2}+8kr+1\) is a perfect square and hence \[ \widetilde{r}=\sqrt{8r^{2}+8kr+1} \] is an integer called upper \(k\)-gap Lucas-balancer corresponding to \(r\). Hence \((n,C)\) is called an upper \(k\)-gap balancing pair and \((r,\widetilde{r})\) is called upper \(k\)-gap balancer pair. Since \(n\) is an upper \(k\)-gap balancing number if and only if \( 8n^{2}+8(1-k)n+(2k-1)^{2}\) is a perfect square, they set \[ y^{2}=8n^{2}+8(1-k)n+(2k-1)^{2} \] for some positive integer \(y\). Then \[ y^{2}=2(2n+1-k)^{2}+2k^{2}-1 \] and taking \(x=2n+1-k\), they get the Pell equation \[ y^{2}-2x^{2}=2k^{2}-1. \] Similarly, since \(r\) is an upper \(k\)-gap balancer if and only if \( 8r^{2}+8kr+1\) is a perfect square, they set \[ y^{2}=8r^{2}+8kr+1 \] for some positive integer \(y\). Then \[ y^{2}=2(2r+k)-2k^{2}+1 \] and taking \(x=2r+k\), they get the Pell equation \[ y^{2}-2x^{2}=-2k^{2}+1. \] They first determined the set of all integer solutions of \(y^{2}-2x^{2}=2k^{2}-1\) and \(y^{2}-2x^{2}=-2k^{2}+1\), and then they determined the general terms of upper \(k\)-gap balancing numbers, upper \(k\)-gap balancers, upper \(k\)-gap Lucas-balancing numbers and upper \(k\)-gap Lucas-balancers. Further the also deduced some algebraic relations on these numbers.