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.