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A set $A$ is MSTD (more-sum-than-difference) or sum-dominant if$|A+A|>|A-A|$, and is RSD (restricted-sum dominant) if $|A\hat{+}A|>|A-A|$,where $A\hat{+}A$ is the set of sums of distinct elements in $A$. We study aninteresting family of MSTD sets that have appeared many times in the literature(see the works of Hegarty, Martin and O'Bryant, and Penman and Wells). Whilethese sets seem at first glance to be ad hoc, looking at them in the right wayreveals a nice common structure. In particular, instead of viewing them asexplicitly written sets, we write them in terms of differences between twoconsecutive numbers in increasing order. We denote this family by $\mathcal{F}$and investigate many of its properties. Using $\mathcal{F}$, we are able togenerate many sets $A$ with high value of $\log|A+A|/\log|A-A|$, construct sets$A$ with a fixed $|A+A|-|A-A|$ more economically than previous authors, andimprove the lower bound on the proportion of RSD subsets of$\{0,1,2,\dots,n-1\}$ to about $10^{-25}$ (the previous best bound was$10^{-37}$). Lastly, by exhaustive computer search, we find six RSD sets withcardinality $15$, which is one lower than the smallest cardinality found todate, and find that $30$ is the smallest diameter of RSD sets.

Generalizations of a Curious Family of MSTD Sets Hidden By Interior Blocks