On balancing and Lucas-balancing numbers expressible as product of two $k$-Fibonacci numbers

Abstract

A positive integer $n$ is called a balancing number if there exists a positive integer $r$ such that $1+2+\cdots +(n-1)=(n+1)+(n+2)+\cdots +(n+r)$. The corresponding value $r$ is known as the balancer of $n$. If $n$ is a balancing number, then $8n^2+1$ is a perfect square, and its positive square root is called a Lucas-balancing number. For any integer $k\ge 2$, let $\{F_n^{(k)}{\}}_{n\ge -(k-2)}$ denote $k$-generalized Fibonacci sequence which starts with $0,\dots ,1$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we investigate all balancing and Lucas-balancing numbers that can be expressed as the product of two $k$-generalized Fibonacci numbers.