For an integer \(k\ge 2\), let \((L_n^{(k)})_n\) be the k-generalized Lucas sequence which starts with \(0,\ldots ,0,2,1\) (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all k-generalized Lucas numbers which are Fibonacci, Pell or Pell–Lucas numbers, i.e., we study the Diophantine equations \(L_n^{(k)}=F_m\), \(L^{(k)}_n = P_m\) and \(L_n^{(k)}=Q_m\) in positive integers n, m, k with \(k \ge 3\).

中文翻译：

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关于k-Lucas序列与一些二元序列的交集

对于整数\(k\ge 2\)，令\((L_n^{(k)})_n\)是k -广义卢卡斯序列，以\(0,\ldots ,0,2,1\ ) ( k项) 之后的每一项都是前面k项的总和。在本文中，我们找到了所有k广义卢卡斯数，即斐波那契数、佩尔数或佩尔-卢卡斯数，即我们研究丢番图方程\(L_n^{(k)}=F_m\) , \(L^{( k)}_n = P_m\)和\(L_n^{(k)}=Q_m\)正整数n ,  m ,  k和\(k \ge 3\)。