I just realized that using both of these:

P_n = n*(3*n - 1)/2
S_n = n*(3*n + 1)/2

is silly:

S_n - P_n = n*(3*n + 1)/2 - n*(3*n - 1)/2 = n

so S_n = P_n + n.

Going another route, if we solve for n via:

S_m = P_n
m*(3*m + 1)/2 = n*(3*n - 1)/2
m*(3*m + 1) = n*(3*n - 1)

Which is trivially true for n = m = 0 and also,

3*m + 1 = 3*n - 1 = 0
n = m + 2/3

So the 2 equations are just translations of each other, as expected. To graph all the 50% thresholds, we would just use the P_n formula.