OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$.

Mikhail Kurkov noticed that it appeared that $a(n+28) = -a(n)$ for this sequence. It's not quite true: the first $n$'s for which this is not true are $578, 1143,$ and $1736$. But still it's remarkably close to true.

A nice illustration of the Strong Law of Small Numbers? Is there any explanation for the almost-identity, or is it just coincidence? Is it still true for large $n$ that $a(n+28)$ is usually $-a(n)$?