A remarkable almost-identity

Question

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 $-a(n)$?

Answer 1

Consider $F(z) = 4/(3+\exp(4z))$ as a function of the complex variable $z$. It is meromorphic and has simple poles where the denominator vanishes. Namely when $4z = \log 3 + (2k +1)\pi i$ for integers $k$. The poles with the smallest magnitude of $z$ occur when $4z = \log 3 \pm \pi i$. We can compute the Taylor series coefficients of $F$ by looking at $$ \frac{1}{2\pi i} \int_{|z|= r} F(z) z^{-n} \frac{dz}{z}, $$ starting with $r$ suitably small. Now we can estimate the integral asymptotically by taking larger values of $r$, and accounting for poles that are encountered. As noted above, the smallest poles are at $(\log 3 \pm \pi i)/4$ and these will account for the leading asymptotics of these coefficients. Now, the argument of $(\log 3 \pm \pi i)/4$ is $\pm 1.23438\ldots $ which is very nearly $11 \pi/28=1.23419\ldots$. This accounts for the observed phenomenon.