The correct relationship between p and P is

1/P(2n)=1/2*(1/p(n)+1/P(n))

and

p(2n)2=p(n)P(2n)

That is, P(2n) is the harmonic average of p(n) and P(n), and p(2n) is the geometric mean of p(n) and P(2n)

This follows from noting that

p(n)= 2 n sin(pi/n)
and
P(n)= 2 n tan(pi/n)

You can use a spreadsheet, calculator, or maple to apply Richardson's extrapolation method. An example of this method for nr can be found here