  The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ of
distinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j
= \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all different
then ${\mathbf x}$ is a linear sequencing and if the partial sums are all
different when $|i-j| \leq t$ then ${\mathbf x}$ is a $t$-weak sequencing. We
investigate these notions of sequenceability in semidirect products using the
polynomial method. We show that every subset of order $k$ of the non-identity
elements of the dihedral group of order $2m$ has a linear sequencing when $k
\leq 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than
$k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$
of a non-abelian group of order three times a prime has a linear sequencing
when $5 < k \leq 10$, unless $s_k$ is unavoidably the identity; and that if the
order of a group is $pe$ then all sufficiently large subsets of the
non-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e \leq
3$ and $t \leq 6$.
