First, we show that the sum-free set generated by the period-doubling sequence is not $κ$-regular for any $κ\geq 2$. 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 $κ$-regular for all $κ\geq 2$. Finally, we study the sum-free sets generated by Sturmian sequences that begin with `11', and their difference sequences.