Problem 353
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, (d/(π r))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 πr 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: (½πr/(πr))2+(½πr/(πr))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 ∑M(2n-1) for 1≤n≤15.
Give your answer rounded to 10 digits behind the decimal point in the form a.bcdefghijk.
危险的月球
月球可以表示为以(0,0,0)为中心、半径为r的球体C(r)。
月球上的站点设立在C(r)表面坐标为整数的点处。位于坐标(0,0,r)的站点称为北极站，位于坐标(0,0,-r)的站点被称为南极站。
所有的站点通过沿大圆圆弧的最短路与其它站点相连。在两个站点之间的旅行是很危险的。如果两个站点之间路径的长度为d，则在两个站点之间旅行的危险程度为(d/(π r))2（或称之为该路径的危险程度）。如果一次旅行包括了多于两个站点，则旅行的危险程度为途中每一段路径的危险程度之和。
直接从北极站到南极站的旅行，其长度为πr，危险程度为1。从北极站到南极站途径(0,r,0)的旅行，总长度不变，但是危险程度更低：(½πr/(πr))2+(½πr/(πr))2=0.5。
从C(r)的北极站到南极站的旅行，其最低危险程度记为M(r)。
已知M(7)=0.1784943998，四舍五入到小数点后10位数字。
对于1≤n≤15，求∑M(2n-1)。
将你的答案四舍五入到小数点后10位数字，即形如a.bcdefghijk。