# Odd-side fills

- 2026-08-30 [Proved] At odd side `n = 2k - 1` the residue split of an axis has `k` low positions and `k - 1` high, so a base-2 flat design fills `sum over its corners of k^(zeros) (k - 1)^(ones)`, and the six designs of the plane read as the polygonal numbers in `k`: low corner `k^2` (A000290), tree `k(2k - 1)` hexagonal (A000384), carpet `k(3k - 2)` octagonal (A000567), void `2k^2 - 2k + 1` centered square (A001844), corner and centre `3k^2 - 3k + 1` centered hexagonal (A003215), solid `(2k - 1)^2` odd squares (A016754); `two_census` at sides 3 to 11 returns `8, 21, 40, 65, 96` for the carpet and `6, 15, 28, 45, 66` for the tree. Witness: mrlymath::formulas::counting fill polynomial, mrlydemo two_census, A000567, A000384.