Function extrapolation
Your data looks logarithmic. Try using the scipy.optimize curve_fit() function to find the approximate log coefficients. I tried several built-in python functions, but couldn't get a good fit on any of them, but they can be used as a starting point.
Finally, I ran the curve_fit() function on the data you posted, and after making a few adjustments, I was able to produce the green line on this chart. The blue line is your original data. They are so close together that it is difficult to see the difference. The yellow line is the one that the curve_fit gave me for a best fit.
I also included the extrapolated points (green line again) and the values for the next 15 points below. As you can see, the line is asymptotic to the x axis.

FYI, My adjustments included adding a np.log() to the denominator to convert the base of the log. Also, I changed the intercept to 1 as that is where your data starts.  If you want to automate this, you could write some functions to iterate over different values of the log base and the intercept. I included comments in the code for more details.
import matplotlib.pyplot as plt  # Matlab-style plotting
import numpy as np
import scipy
from scipy.optimize import curve_fit
x = [1.0,2.0,3.0,4.0,5.0,6.0]
x2 = np.arange(0, 22.0, 1.0)
y = [1.0, 0.488, 0.300, 0.213, 0.163, 0.127]
x = np.array(x)
y = np.array(y)

plt.plot(x,y)
# get the approximate values for the log function
data = scipy.optimize.curve_fit(lambda t,a,b: a+b*np.log(t),  x,  y, p0=(1, 1))
# plot those values, the fit is not very good
y1 = data[0][0] + data[0][1]*np.log(x) # yellow
plt.plot(x,y1)
# original coefficients were .91 and -.48
# adjust the 2 coefficents until you get a better fit
# first coefficient is like the intercept but for the line of x of 1, not the y axis
# I also changed the log base value by including the np.log() on the denominator, 
# this gives the line a larger curve than the original
y2 = 1.0 - np.log(x2)/np.log(x2+1.9) # green
plt.plot(x2,y2)
plt.show

data

(array([ 0.91083753, -0.48242954]),
 array([[ 0.00543936, -0.00380311],
        [-0.00380311,  0.00346828]]))

y2

array([       inf, 1.        , 0.49069866, 0.30871637, 0.21896813,
       0.16675119, 0.13310187, 0.10985155, 0.09295092, 0.08018302,
       0.0702405 , 0.06230656, 0.05584666, 0.05049745, 0.04600401,
       0.04218259, 0.03889766, 0.03604721, 0.03355312, 0.03135462,
       0.02940378, 0.02766231])