A395434 Minimum number of hexagonal cells in a connected polyhex that contains every one-sided n-hex as a rotation-and-translation subset.

1, 2, 4, 7, 11, 15, 21

Offset

1,2

Comments

A one-sided n-hex is a connected set of n unit hexagonal cells on the hex lattice, counted up to rotation only; reflections are distinct pieces. The one-sided polyhex count is A006535.

a(n) is the minimum number of cells in a connected polyhex that contains every one-sided n-hex under some rotation and translation of the piece.

Each reported value a(n) for n = 1..7 is proved by SAT search inside an (n+1) X (n+1) axial rectangle with a machine-verified UNSAT proof at size a(n) - 1 (drat-trim "s DERIVATION" for n = 1..6; cadical --lrat + lrat-check "c VERIFIED" for n = 7). Every value is also cross-checked by an independent pure-Python geometric containment verifier with a disjoint code path from the solver.

The first seven terms coincide with A293239, A261878, A261993, and A299251, which are defined in unrelated domains (derivatives of x^x, fractional-part number theory, divisor-sum formula). At the index aligned with A395434(8), each of these matchers takes the value 28: A293239(7) = 28, A261878(8) = 28, A261993(8) = 28, A299251(10) = 28. A 26-cell connected polyhex containing all 2821 one-sided 8-hexes has been constructed and independently verified by a geometric containment check on every piece (pure-Python enumeration via A006535, no SAT), giving the upper bound a(8) <= 26 < 28. Therefore A395434 differs from each of A293239, A261878, A261993, A299251 at n = 8. See the GitHub repo for the 26-cell construction, the verifier, and the b-file audit.

Links

Table of n, a(n) for n=1..7.

Peter Exley, Solver code, verifiers, DRAT/LRAT certificates, figures, GitHub.

Example

For n = 1, a(1) = 1: a single hexagonal cell contains the unique one-sided 1-hex.
For n = 2, a(2) = 2: a 2-cell domino contains the unique one-sided 2-hex.
For n = 3, a(3) = 4: a 4-cell connected polyhex with bounding box 2 X 3 contains all 3 one-sided 3-hexes.
For n = 7, a(7) = 21: a 21-cell connected polyhex with bounding box 8 X 5 contains all 620 one-sided 7-hexes under rotation + translation.

Crossrefs

Cf. A006535 (number of one-sided n-hexes), A000228 (number of free n-hexes), A001207 (number of fixed n-hexes), A327094 (square-grid analog, free polyominoes), A392363 (triangular-grid analog, free polyiamonds), A395422 (triangular-grid analog, fixed polyiamonds).

Sequence in context: A077169 A094277 A263995 * A389319 A293239 A261878

Adjacent sequences: A395430 A395431 A395432 * A395435 A395437 A395438

Keyword

nonn,more

Author

Peter Exley, Apr 24 2026

Status

approved