Problem 645

Every Day is a Holiday

On planet J, a year lasts for $D$ days. Holidays are defined by the two following rules.

At the beginning of the reign of the current Emperor, his birthday is declared a holiday from that year onwards.
If both the day before and after a day $d$ are holidays, then $d$ also becomes a holiday.

Initially there are no holidays. Let $E(D)$ be the expected number of Emperors to reign before all the days of the year are holidays, assuming that their birthdays are independent and uniformly distributed throughout the $D$ days of the year.

You are given $E(2)=1$, $E(5)=31/6$, $E(365)\approx 1174.3501$.

Find $E(10000)$. Give your answer rounded to $4$ digits after the decimal point.

每天都是休息天

J星球上的一年有$D$天，并按照下面两条规则安排休息天：

每当一名新皇帝登基后，他的生日被宣告为一个新的休息天；
如果一年中第$d$天的前一天和后一天都是休息天，则这一天自动成为休息天。

一开始，这个星球上没有休息天；后来随着一名又一名新皇帝登基，每一天都变成了休息天。假设皇帝们的生日独立地均匀分布在一年中的每一天，并记$E(D)$为恰好每一天都变成休息天时登基过的皇帝总数的期望。

已知$E(2)=1$，$E(5)=31/6$，$E(365)\approx 1174.3501$。

求$E(10000)$，并将答案保留小数点后$4$位小数。