A396101
live on OEIS
Number of fixed polycubes with n cells that have cavities
New sequence. All eight known terms, n = 11 through 18. Author: Stephen J. Pursey, Aug 10 2026.
+8 terms
Aug 10 2026
The ledger
Contributions
Sequences authored, extended, and corrected; results proven along the way.
Live on OEIS — published, permanent
A357083
extendedlive on OEIS
Number of free polycubes with n cells that contain holes
a(15)–a(18) added: 422277, 4310738, 41982903, 395335115. Four terms no one had computed.
+4 terms
Aug 10 2026
A355966
correctedextendedlive on OEIS
Number of polycubes with n cells that have at least one hole
a(14) = 76017 and a(15) = 838575 corrected — the published values had stood wrong since 2022. a(16)–a(18) added. The faulty source program was removed from the record after a two-bug autopsy accounting for its deficits to the exact digit.
+3 terms
2 corrected
Aug 11 2026
In editorial review
A060966
correctedextendedin review
Number of non-isomorphic oriented circulant graphs of order n
a(8) = 9, a(12) = 70, a(15) = 290 corrected — errors traced 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. Triple-verified with independent isomorphism engines.
+5 terms
3 corrected
Aug 11 2026
Proven along the way
Thm
proven
The smallest polycube hiding a two-cell void has 15 cells
Minimality proven, and the census taken: exactly 4320 fixed examples, 180 one-sided. Both numbers are new.
Aug 10 2026
octahedron-384
proven
a(11) = 384 = the number of spanning trees of the octahedron
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.
Aug 10 2026
Thm
proven
Exactness lemmas for threshold pruning
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.
Aug 10 2026
error-autopsies
correctedproven
Two published artifacts diagnosed to the exact digit
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.
Aug 11 2026
oriented-ci
proven
Z8, Z9, Z18 fail the Cayley isomorphism property for oriented graphs
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.
Aug 11 2026
two-cavity-minimum
proven
The smallest polycube with two cavities has exactly 17 cells
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.
Aug 11 2026