Skipping squares revisited

For positive integer $k$ ($k>1$), we begin a sum $s$ at 0 and repeatedly apply a process: we add one integer to $s$ from 1 to $k$ with equal probability. The process ends when $s$ is a perfect square. For example, if $s$ goes through 0, 2, 3, 5, 7, 9, the process ends at $s$ = 9, and two squares 1 and 4 were skipped over.

Let $E(k)$ be the expected number of perfect squares skipped over when the process finishes for parameter $k$. For example, $E(2)=0.6832612735$ and $E(10)=4.9658004390$, both rounded to 10 digits after the decimal point.

Find $E(100)$. Give your answer rounded to 10 digits after the decimal point.

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This problem was inspired by Problem 648 at Project Euler (https://projecteuler.net/problem=648)