A Simple Regression Problem

This article is part of a new series featuring problems with solution, to help you hone your machine learning and pattern recognition skills. Try to solve this problem by yourself first, before looking at the solution. Today's problem also has an intriguing mathematical appeal and solution: this allows you to check if your solution found using machine learning techniques, is correct or not. The level is for beginners. 

The problem is as follows. Let X1, X2, X3 and so on be a sequence recursively defined by Xn+1 = Stdev(X1, ..., Xn). Here X1, the initial condition, is a positive real number or random variable. Thus,

It is clear that Xn = An X1, where An is a number that does not depend on X1. So we can assume, without loss of generality, that X1 = 1. For instance, A1 = 1 and A2 = 0. The purpose here is to study the behavior of An (for large n) using simple model fitting techniques. I plotted the first few values of An, below. In the figure below, the X-axis represents n, and the Y-axis represents An. The question is: how to approximate An as a simple function of n? Of course, a linear regression won't work. What about a polynomial regression?

Read the full solution here.