John Faulkner, Sr. who came to America in 1655 had a son named Francis 1648-1786. Last nite I found another Francis: Lt. Francis Faulkner 1651-1732. His father was Edmund Faulkner who immigrated to America in 1639. Edmund's parents were Richard Faulkner 1590-1670 and Joan Maidunk 1595-1650. Lt. Francis Faulkner is the Francis who married Abigail Dane. Abigail has her own page on Wikipedia.
So, my search has taken a turn. I'm still researching the Faulkners of Massachusettes. I still intend to go back to Nathan Faulkner born about 1745 and the father of Dr. Archibald Faulkner.