rosettes

call 'rosettes'
a number of circles that overlap as much as is possible...
with n (circles) = 3,
r (regions) = 7

n = 4
r = 13

n = 8
r = 57
















in general?
why?

if you focus your attention on the 'swirls' then each of the 8 of them has 7 regions creating
8 x 7 + 1  (the middle) altogether