Markoff–Rosenberger triples and generalized Lucas sequences

Abstract We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ a x 2 + b y c z = d with $$(x,y,z)=(U_i,U_j,U_k)$$ ( , ) U i j k , where $$U_i$$ denotes i -th generalized Lucas number of first/second kind. provide an upper bound for minimum indices and we apply result to completely resolve concrete equations, e.g. determine solutions containing only balancing numbers Jacobsthal numbers, respectively.