Closed-form enumeration of polyominoes

Determine a closed-form expression for the total number of free polyominoes of size n for general n.

Background

The paper studies the enumeration of free polyominoes, in which shapes related by rotation or reflection are identified. Although exact values are known for small sizes and the paper extends numerical enumeration through n=59, the general enumeration problem remains unresolved. A closed-form formula would characterize the sequence Free(n) beyond individually computed values and provide a theoretical solution to the longstanding counting problem.

References

The problem of enumerating $n$-ominoes is one of the classic and unsolved problems in combinatorics. While the exact value of $\mathrm{Free}(n)$ is known for small $n$, a closed-form expression for general $n$ has not yet been discovered.

— Enumeration of Polyominoes up to Size N=59  (2510.22446 - Shirakawa, 25 Oct 2025) in Section Introduction