A Question on Name Suffixes?

Question

If a father was named John Smith Sr and his son was named John Smith Jr, and then the son had a son named John Smith what would the suffix for the son's son be?

Would the first father become John Smith I and the first son John Smith II and the son's son John Smith III? Or would the the Sr. stay a Sr and the Jr stay a Jr?

Answer 1

*****My grandfather was William Moore (no senior) since there was no idea when he was born that there would be a junior.  My father was born and was William Moore, Jr.  My brother was born and was William Moore, III.

Now in the early 1800's I find that there were two Philip Smiths in the same town.  One was Sr., one was Jr.  They were NOT related, however.  It was merely a designation to distinguish the elder from the younger, not necessarily family members from each other.

Answer 2

The son of John Smith Jr. would be John Smith the 3ed. ,if he had a son he would be John Smith the 4th.Here is one for you that I came across. [1] John Barrett Sr,married Lucy Barney ,then the records say that Lucy Barrett and John Barrett [2] had John Barrett Jr.[3] ,As crazy as this looks it fit right in what family oral history had always said ,that John Barrett Sr. died young and Lucy Barrett Married a Native American That took over her dead husbands ID.[because Native people were not allowed to own land in this time period She  allowed him to do this] and there son was John Barrett Jr.[3]