  We study how to generate binary de Bruijn sequences efficiently from the
class of simple linear feedback shift registers with feedback function $f(x_0,
x_1, \ldots, x_{n-1}) = x_0 + x_1 + x_{n-1}$ for $n \geq 3$, using the cycle
joining method. Based on the properties of this class of LFSRs, we propose two
new generic successor rules, each of which produces at least $2^{n-3}$ de
Bruijn sequences. These two classes build upon a framework proposed by Gabric,
Sawada, Williams and Wong in Discrete Mathematics vol. 341, no. 11, pp.
2977--2987, November 2018. Here we introduce new useful choices for the
uniquely determined state in each cycle to devise valid successor rules. These
choices significantly increase the number of de Bruijn sequences that can be
generated. In each class, the next bit costs $O(n)$ time and $O(n)$ space for a
fixed $n$.
