Integers representable by \((x+y+z)^3/xyz\) (Q1380964)

\textit{A. Bremner} and \textit{R. K. Guy} [Proc. Edinb. Math. Soc., II. Ser. 40, 1-17 (1997; Zbl 0874.11032)] discuss the problem of finding integers which may be represented by \((x+y +z)^3/xyz\), where \(x,y,z\) are integers. To this end, they present tables of solutions for integers \(n\) in the range \(-200\leq n\leq 200\) and offer several parametric solutions which involve both positive and negative integers. Here four infinite families of solutions which involve only positive integers are presented. Furthermore, these families contain sequences that are generated by linearly recursive relations.