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。