Problem 469
Empty chairs
In a room N chairs are placed around a round table.
Knights enter the room one by one and choose at random an available empty chair.
To have enough elbow room the knights always leave at least one empty chair between each other.
When there aren’t any suitable chairs left, the fraction C of empty chairs is determined.
We also define E(N) as the expected value of C.
We can verify that E(4) = 1/2 and E(6) = 5/9.
Find E(1018). Give your answer rounded to fourteen decimal places in the form 0.abcdefghijklmn.
空椅子
在一间屋子中，围绕圆桌摆了N把椅子。
骑士们依次进入屋子，并且随机选择一把空椅子坐下。
为了保证手肘不相互撞击，相邻的骑士之间至少要空出一张空椅子。
当剩下的空椅子都不能坐时，可以计算出空椅子的比例C。
记E(N)是C的期望值。
可以验证，E(4) = 1/2，而E(6) = 5/9。
求E(1018)，给出其小数点后14位数字，即0.abcdefghijklmn。