Nonsequenceable Steiner triple systems

A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0\le a\le (n-1)\mathrm{/}3$, then there exists a nonsequenceable PSTS$(n)$ of size $\frac{1}{3}\left(\genfrac{}{}{0px}{}{n}{2}\right)-a$, for all $n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}6)$ except for $n=7$.