On \( b\)-concatenations of two \( k\)-generalized Fibonacci numbers

Abstract

Let \( k \geq 2 \) be an integer. One of the generalization of the classical Fibonacci sequence is 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 \( F_{1}^{(k)}=1\) \(. F_{n}^{(k)} \) is an order \( k \) generalization of the Fibonacci sequence and it is called \( k\)-generalized Fibonacci sequence or shortly \( k\)-Fibonacci sequence. Banks and Luca [7], among other things, determined all Fibonacci numbers which are concatenations of two Fibonacci numbers. In this paper, we consider the analogue of this problem in more general manner by taking into account the concatenations of two terms of the same sequence in base \(b \geq 2\). First, we show that there exists only finitely many such concatenations for each \( k \geq 2 \) and \( b \geq 2 \). Next, we completely determine all these concatenations for all \( k \geq 2\) and \( 2 \leq b \leq 10 \).