Risky Moon

A moon could be described by the sphere $C(r)$ with centre $(0,0,0)$ and radius $r$.

There are stations on the moon at the points on the surface of $C(r)$ with integer coordinates. The station at $(0,0,r)$ is called North Pole station, the station at $(0,0,-r)$ is called South Pole station.

All stations are connected with each other via the shortest road on the great arc through the stations. A journey between two stations is risky. If d is the length of the road between two stations, $\left(\frac{d}{\pi r}\right)^2$ is a measure for the risk of the journey (let us call it the risk of the road). If the journey includes more than two stations, the risk of the journey is the sum of risks of the used roads.

A direct journey from the North Pole station to the South Pole station has the length $\pi r$ and risk $1$. The journey from the North Pole station to the South Pole station via $(0,r,0)$ has the same length, but a smaller risk:
\[
\left(\frac{\frac{1}{2}\pi r}{\pi r}\right)^2+\left(\frac{\frac{1}{2}\pi r}{\pi r}\right)^2=0.5
\]

The minimal risk of a journey from the North Pole station to the South Pole station on $C(r)$ is $M(r)$.

You are given that $M(7)=0.1784943998$ rounded to $10$ digits behind the decimal point.

Find $\displaystyle{\sum_{n=1}^{15}M(2^n-1)}$.

Give your answer rounded to $10$ digits behind the decimal point in the form a.bcdefghijk.

Firstly, we note that the journey from the North Pole station to the South Pole station will go through the origin. Since the distance from every station from the origin is `r`, it will not be necessary to find out the specific coordinates of the stations. Hence, let’s denote the `(i, j, k)` station by `(p,q)`, where `p` is the number of integer steps taken towards `(1,0,0)` after reaching the origin from `(0,0,r)` and `q` is the number of integer steps taken towards `(0,1,0)` after reaching the origin from `(0,0,r)`. Initially we start from the `(0,0,r)` station denoted by `(0,0)`.

Now, the total risk of the journey from `(0,0)` to `(p,q)` is `(1-p/r)^2 + (1-q/r)^2`. Let’s call this `R(p,q)`. Note that `R(p,q)` is a monotonically decreasing function of `p` and `q`. The station `(p,q)` can be reached from either `(p-1,q)` or `(p,q-1)`.

Hence, we have `R(p,q) = min {R(p-1,q), R(p,q-1)}`.

We compute `R(p,q)` for all `p` and `q` less than or equal to `n` and store the minimum risk paths. By nature of the monotonicity of `R(p,q)`, we only need to look at the stations `(p=p1,q=r-p1) for p1 <= r/2`. After having computed the `M(r)`, we simply sum them over the given range of `n` and round off our answer to 10 decimal places. I would like to note that to actually solve this problem, considering this is a challenging problem, numerical methods should be applied with precise algorithms. Python or any similar programming languages should be applied to solve this. This should give a solution to this in a pragmatic way. However, I can guide you through writing a code to solve this problem upon request. Unfortunately, I cannot compose the result without using a software to accurately calculate the sum up to the 15th term. This method or approach is outlined for understanding purpose and it should give you the proper understanding of the approach required to solve this problem. Theoretical mathematics is applied in this way to solve the problems with precise answers.

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