I have been wondering how many ancestors I might have...

Two parents, four grandparents, eight great grandparents... If I continue this series back at 3 generations per century, I find I have over a billion Viking-era ancestors. And 10^18 in the Roman era. And so on. This is plainly wrong. Where is the flaw in the logic?

I am told the flaw's this: I've presumed each couple has only one child, so a pyramid is formed which expands in size as you go backwards. Actually, people had on average many more than two children for the last 2000+ years, so as the groups intermarry, grandparenting for 12 or 20 means there's duplication in the figures. I'm counting child-grandparent LINKS not the actual number of people: 2 grandparents with 20 grandkids would show up as 20 by my counting. I may have 10^18 ancestor-links, but due to cross-breeding (often miles apart on the family tree), that's fewer actual PEOPLE by far. Easy example: g/g/parents have two kids, both marry, one offspring from each marriage marry together: my parents. Therefore I have 2p, 4gp, 6ggp - not 8ggp. And so on. More duplication as I go up.