For this problem, the orientation of a protein is important; e.g. HPP is considered distinct from PPH. Thus, there are $2^n$ distinct proteins consisting of $n$ elements.
When one encounters these strings in nature, they are always folded in such a way that the number of H-H contact points is as large as possible, since this is energetically advantageous.
As a result, the H-elements tend to accumulate in the inner part, with the P-elements on the outside.
Natural proteins are folded in three dimensions of course, but we will only consider protein folding in <u>two dimensions</u>.
<imgclass="img-responsive center-block"alt="two possible ways to fold example protein"src="https://cdn.freecodecamp.org/curriculum/project-euler/protein-folding.gif"style="background-color: white; padding: 10px;">
The folding on the left has only six H-H contact points, thus it would never occur naturally. On the other hand, the folding on the right has nine H-H contact points, which is optimal for this string.
Assuming that H and P elements are equally likely to occur in any position along the string, the average number of H-H contact points in an optimal folding of a random protein string of length 8 turns out to be $\frac{850}{2^8} = 3.3203125$.
What is the average number of H-H contact points in an optimal folding of a random protein string of length 15? Give your answer using as many decimal places as necessary for an exact result.