--description--

In a very simplified form, we can consider proteins as strings consisting of hydrophobic (H) and polar (P) elements, e.g. HHPPHHHPHHPH.

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>.

The figure below shows two possible ways that our example protein could be folded (H-H contact points are shown with red dots).

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.

--hints--

proteinFolding() should return 8.0540771484375.

assert.strictEqual(proteinFolding(), 8.0540771484375);

--seed--

--seed-contents--

function proteinFolding() {

  return true;
}

proteinFolding();

--solutions--

// solution required