Into How Many Regions Can Six Planes Divide Space?

I recently misread an easy recreational puzzle as 'Into how many regions can six planes divide space?'.

It took me about half an hour to get an answer. (Which was the correct answer to the question, but not the answer to the real puzzle). But it turned out to be great fun to think about.

I'm pretty sure that my answer's correct (and wikipedia agrees), but although the combination of mathematical proof and worldwide consensus should probably be enough, I'm never really convinced by a mathematical idea until I've seen it confirmed by lots of examples.

So I've got programs planned, and if they turn out well, then I'll post them here. But they won't be any fun to read unless you've tried to work it out yourself.

If it's not clear, the idea is that you've got a big block of sponge cake, and a very large knife, and you want to cut it into as many pieces as possible with six cuts.

So if the problem said 'one' rather than 'six', the answer would be two, and if it said 'two', then the answer would be 'four'.

1,2,4,8,..... how does this sequence continue?