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Let $\bar{S}$ denote the set of integers $n$ such that $n!$ cannot be writtenas a sum of three squares. Let $\bar{S}(n)$ denote $\bar{S} \cap [1, n]$. Weestablish an exact formula for $\bar{S}(2^k)$ and show that $\bar{S}(n) = 1/8*n+ \mathcal{O}(\sqrt{n})$. We also list the lengths of gaps appearing in$\bar{S}$. We make use of the software package Walnut to establish theseresults.

Factorials and Legendre's three-square theorem: II