The ledger
Sequences authored, extended, and corrected; results proven along the way.
New sequence: 11, 15, 19, 21, 24, 26, 28, 29, 32, 34, 36, 37, 39. Each term is a minimum over every possible cavity shape of the forced wall plus the cheapest way to connect it, computed by two independently written programs that agree at every n. The three-dimensional version of A283056. Author: Stephen J. Pursey, Aug 19 2026.
New sequence: 1, 2, 6, 23, 115, 688, 4795, 37145, 310233. It began as a comment proposed for A338660, that these diagrams are counted by the open meander numbers; a reviewer found the comment false at n = 2, and the true counts were in no entry. Author: Stephen J. Pursey, Aug 26 2026.
a(7)–a(9) added: 802, 4630, 30678, the first new terms since the entry was created in 2021. Two independent graph-isomorphism engines agree on every term, and the number of curves generated matches the open meander numbers A005316(2n) at every size.
a(11)–a(17) added, the first new terms since 2014; a(17) = 6324511917177082209. A transfer-matrix method replaces brute force, and independently written implementations agree on every term.
a(17) = 1332348825838 added, the first new term since 2008. Two differently designed programs agree.
New sequence. All eight known terms, n = 11 through 18. Author: Stephen J. Pursey, Aug 10 2026. Sep 13: proposed a comment recording the 18-cell two-cavity count, 590074.
a(15)–a(18) added: 422277, 4310738, 41982903, 395335115. Four terms no one had computed.
a(14) = 76017 and a(15) = 838575 corrected; the published values had been wrong since 2022. a(16)–a(18) added. The faulty source program was removed from the record after a two-bug autopsy that accounted for its deficits to the exact digit.
a(8) = 9, a(12) = 70, a(15) = 290 corrected; the errors trace to Table 1 of the source paper. a(16)–a(20) added. Resolves a discrepancy flagged in the entry since 2017 and confirms A283189 (Howroyd) correct at all n ≤ 20. Verified three separate ways with independent isomorphism engines. Published Sep 3 after editing by Andrew Howroyd.
Companion to A399316 in which a diagram and its mirror image count separately: 1, 2, 6, 28, 166, 1141, 8645, 70351, 603253. Split out at an editor's request. Proposed Sep 13.
Brendan McKay asked in 2010 where the term a(8) = 71 came from. Under the stated definition a(8) = 51, the same as A000512, and McKay's suggested explanation, matrices with sorted rows and columns, already disagrees with the entry at n = 5. The editors are deciding whether to retire the entry as a duplicate.
Rejected Aug 31 and the number recycled. The editors want at least four nonzero terms before accepting the sequence, and two are known: a(17) = 16968 and a(18) = 590074, computed for the review. a(17) is recorded in A396101 and a(18) has been proposed there. The next terms need more memory than this machine has.
A polycube in d dimensions with a cavity has at least 4d − 1 cells, and the minimal ones are exactly the spanning trees of the cross-polytope graph: (2d)^(d−2)(2d−2)^d of them, 4, 384, 82944, 32768000. Up to symmetry they are the nets of the d-cube: 1, 11, 261, 9694. Proof in the linked note; checked by exhaustive search through four dimensions. Comments recording this interpretation in A193130 and A091159 are prepared and waiting for a free submission slot.
Minimality proven, and the census taken: exactly 4320 fixed examples, 180 one-sided. Both numbers are new.
The first term of A396101 has a closed form: every minimal cavity polycube is six forced sealers plus five connectors, and the connectors are counted by Kirchhoff's theorem on the octahedron graph.
Leafy-cavity lemma; min-degree-3 cavities need at least 24 cells; leafless cavities need at least 19 shell cells up to size 17, exhaustively; a degree-free tail bound. Together these make the enumeration provably exact, not heuristic.
The A355966 source program: two independent bugs (a modeling gap and an unsound pruning step) accounting for every deficit exactly. The A060966 source paper: an odd-p formula hypothesis explaining Table 1's errors.
Empirical: the three exceptional orders of the undirected CI classification all fail CI in the oriented case. Literature check in progress to establish whether this is known.
Exhaustive census at sizes <= 16 (no two-cavity polycubes; no 3+-cell cavities; domino-cavity census at 16: 127,416 fixed / 5,328 one-sided) combined with explicit verified 17-cell constructions.
An exhaustive scan over every polycube shape up to 17 cells confirms no cavity can slip past a two-signature test at 19 cells or fewer. The pruning that makes the count roughly fifteen times faster is proven correct, with zero counterexamples across the whole search.
Answers three questions from Knuth's 2020 note. K3,3 and the third prism marking are unrealizable; the eight-corner catalog is complete once a missing marking of the lopped prism is added, giving thirty in all; and the smallest nonplanar skeleton has exactly fourteen corners, as he conjectured. Every solid carries an exact rational certificate.