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Every Day is a Holiday

Problem 645 Published on Sunday, 2nd December 2018, 01:00 am; Solved by 69

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

  1. At the beginning of the reign of the current Emperor, his birthday is declared a holiday from that year onwards.
  2. 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.