Another late reply -- I'm also a Billington descendant; John is my 10th great-grandfather, via Francis's daughter Mercy.
My great-aunt always thought we might have Mayflower passengers, and was always trying to find them; I'm not sure she expected the Billingtons, though!
Francis Billington m Christian Penn, Plymouth, MA
Mercy (Billington) Martin m John Martin, Rehoboth, MA
Desire (Martin) Carpenter m. Jotham Carpenter, Rehoboth, MA
Jotham Carpenter (jr) m Mehitable Thompson, Rehoboth, MA
Dorcas Carpenter m. James Bullock, Rehoboth, MA
Oliver Bullock m. Susannah Tisdale, Rehoboth, MA
And their son James Sullivan Bullock married Nancy Barrows, and moved (with most of his siblings) from Rehoboth Massachusetts to Deer Park Illinois, where their daughter, Sarah Amelia Dickinson, met a Hadley Dickinson, Henry Anderson Dickinson, and they married and moved back to Western Massachusetts.
and their son, Homer Ferry Dickinson, was my great-grandfather.