On Concatenations of Two $ k $-Generalized Fibonacci Numbers

Abstract

Let $ k \geq 2 $ be an integer. The $ k- $generalized Fibonacci sequence is a sequence defined by the recurrence relation $ F_{n}^{(k)}=F_{n-1}^{(k)} + \cdots + F_{n-k}^{(k)}$ for all $ n \geq 2$ with the initial values $ F_{i}^{(k)}=0 $ for $ i=2-k, \ldots, 0 $ and $ F_{1}^{(k)}=1.$ In 2020, Banks and Luca, among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem by taking into account $ k-$generalized Fibonacci numbers as concatenations of two terms of the same sequence. We completely solve this problem for all $ k \geq 3.