Doubly Sequenceable Groups

Given a sequence $\mathbf{g}:g_0,\dots ,g_m$, in a finite group $G$ with $g_0=1_G$, let $\bar{\mathbf{g}}:{\bar{g}}_0,\dots ,{\bar{g}}_m$, be the sequence defined by ${\bar{g}}_0=1_G$ and ${\bar{g}}_i=g_{i-1}^{-1}{g}_i$ for $1\le i\le m$. We say that $G$ is doubly sequenceable if there exists a sequence $\mathbf{g}$ in $G$ such that every element of $G$ appears exactly twice in each of $\mathbf{g}$ and $\bar{\mathbf{g}}$. If a group $G$ is abelian, odd, sequenceable, R-sequenceable, or terraceable, then $G$ is doubly sequenceable. In this paper, we show that if $N$ is an odd or sequenceable group and $H$ is an abelian group, then $N\times H$ is doubly sequenceable.