A class of Fibonacci-type sequences (Q1846450)

Let \(\{L_n : n\ge 1\}\) be a sequence of the form \[ L_n= \min\left( \sum_{j=1}^p L_{n-a_j}\quad (n>e),\quad \sum_{j=1}^q L_{n-b_j}\quad (n>e)\right), \] where \(\{a_j\}\) and \(\{b_j\}\) are positive integers, and \(e = \max_{i,j} \{a_i ,b_j\}\). A necessary and sufficient condition on the integers \(\{a_j\}\) and \(\{b_j\}\) is given so that, for all choices of positive initial values \(L_1, L_2,\ldots, L_n\), \(L_n = \sum_{j=1}^p L_{n-a_j}\) for all large enough \(n\).