In my work with Pat(2n+1,n), I've come across an interesting relationship.

(What's Pat(2n+1,n)?  see:
http://www.rankyouragent.com/primes/primes_simple.htm
or
http://www.rankyouragent.com/primes/primes.htm)

I have developed formulae which predict the number of constellations between P(n) and P(n)^2.
I've tested the correlation up to P(400).

Here are the formulae:

DEFINE P(N): P(1) = 3, P(2) = 5, P(3) = 7, P(4) = 11, ...
// ie, all the primes, starting with 3

DEFINE P(N)# = 2 * P(1) * P(2) * ... * P(n)
// ie, the primorial, all the primes up to P(n) multiplied together (including 2)

DEFINE #Primes[P(n)->P(n)^2] = ((primeCandidates(n-1) * (P(n)^2-P(n))) / P(n)#) - (0.145348*n*log10(n))^2

// p, p+2
DEFINE #Twins[P(n)->P(n)^2] = ((twinCandidates(n-1) * (P(n)^2-P(n))) / P(n)#) - (0.058652*n*log10(n))^2

// p, p+2, p+6
DEFINE #Triplets[P(n)->P(n)^2] = ((tripletCandidates(n-1) * (P(n)^2-P(n))) / P(n)#) - (0.02881*n*log10(n))^2

// p, p+2, p+6, p+8
DEFINE #Quadruplets[P(n)->P(n)^2] = ((quadrupletCandidates(n-1) * (P(n)^2-P(n))) / P(n)#) - (0.00718*n*log10(n))^2

// primeCandidates are 0's in Pat(2n+1,n) which may or may not become primes
DEFINE primeCandidates(n) = primeCandidates(n-1) * (P(n)-1)  AND
       primeCandidates(1) = 2

// twinCandidates are 00's in Pat(2n+1,n) which may or may not become prime twins
DEFINE twinCandidates(n) = twinCandidates(n-1) * (P(n)-2)  AND
       twinCandidates(1) = 1

// tripletCandidates are 0010's in Pat(2n+1,n) which may or may not become prime triplets
DEFINE tripletCandidates(n) = tripletCandidates(n-1) * (P(n)-3)  AND
       tripletCandidates(2) = 2

// quadrupletCandidates are 00100's in Pat(2n+1,n) which may or may not become prime quadruplets
DEFINE quadrupletCandidates(n) = quadrupletCandidates(n-1) * (P(n)-4)  AND
       quadrupletCandidates(2) = 1

The results can be viewed here:
http://www.rankyouragent.com/primes/twins.xls
(be sure to view all 4 worksheets)

The code that generated it can be found here:
http://www.rankyouragent.com/primes/twins.html