OEIS A000104: Free polyominoes without holes

We explore the counting of free polyominoes without holes—the distinct edge-to-edge shapes formed by n squares, up to symmetry. From the early terms 1, 1, 1, 2, 5, 12, 35 to the rapid combinatorial explosion, the problem becomes increasingly hard and the case for n = 29 remains open. We’ll discuss how researchers count these shapes, the algorithms and computational limits, and why these simple shapes connect to chemistry, tiling, and data packing—showing how the OEIS reveals deep questions behind a seemingly simple idea.

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