  First, we show that the sum-free set generated by the period-doubling
sequence is not $\kappa$-regular for any $\kappa\geq 2$. Next, we introduce a
generalization of the period-doubling sequence, which we call the
period-$k$-folding sequences. We show that the sum-free sets generated by the
period-$k$-folding sequences also fail to be $\kappa$-regular for all
$\kappa\geq 2$. Finally, we study the sum-free sets generated by Sturmian
sequences that begin with `11', and their difference sequences.
