A strange occurrence in the decimal digits of $\pi$?
This is a known property of the Leibniz–Gregory series, and has been used to actually compute $\pi$ to many digits using this series. It arises from the Euler–Maclaurin formula: $$\frac{\pi}{2} - 2 \sum_{k=1}^\frac{N}{2} \frac{(-1)^{k-1}}{2k-1} \sim \sum_{m=0}^\infty \frac{E_{2m}}{N^{2m+1}}$$ where $E_n$ are the Euler numbers. When $N$ (the number of terms) factors into twos and fives only (such as powers of $10$), the error (last sum) becomes a sum of finite decimals.