  Recently, Matsuhira, Matsusaka, and Tsuchida revisited old studies on the
integrality of $k$-G\"{o}bel sequences and showed that the first 19 terms are
always integers for any integer $k\ge 2$. In this article, we further explore
two topics: Ibstedt's $(k,l)$-G\"{o}bel sequences and Zagier's asymptotic
formula for the $2$-G\"{o}bel sequence, and extend their results.
