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%I #10 Apr 12 2026 17:22:43
%S 1,2,3,5,7,10,13,16,19,24,27,32,37,42
%N Smallest connected polyiamond containing all free n-iamonds under rigid motions of the triangular lattice.
%C A polyiamond is a connected set of unit equilateral triangles (cells) in the triangular lattice, joined edge-to-edge. Each cell has exactly 3 edge-neighbors. Free n-iamonds are equivalence classes under the full symmetry group of the lattice (the dihedral group D_6 together with all lattice translations); the count of free n-iamonds is A000577(n). A placement of a free n-iamond into a container is a D_6 orientation followed by a lattice translation (dr, dc) with (dr+dc) even; the parity condition ensures the translation maps the tiling onto itself, preserving cell orientation.
%C Triangular-grid analog of A327094 (square grid). For n >= 4, a(n) is smaller than the square-grid values, consistent with the lower connectivity of the triangular lattice (3 edge-neighbors per cell vs 4 for squares).
%C Conjecture: lim_{n -> infinity} a(n)/n^2 = 1/5. The ratio a(n)/n^2 decreases from 0.240 at n=10 to 0.214 at n=14 (monotonic in this range). For all n <= 14, a(n) <= ceiling((n^2+2n)/5), with equality for n <= 8 and n=10, and a(n) is 1-3 less for n=9,11,12,13,14.
%C The optimal containers for n=11 and n=14 have non-contiguous rows: the bottom row consists of isolated cells at scattered positions (e.g., columns 2, 6, 10 for the n=14 container). Row-contiguity heuristics are therefore unsafe for computing this sequence.
%C a(1)-a(14) were proved using an unconstrained SAT solver (Glucose 4.2 via PySAT) with lazy connectivity cuts (CEGAR). For each n, the solver found a satisfying assignment at k = a(n) and proved unsatisfiability at k = a(n)-1, establishing optimality. Every reported container was additionally verified by two independent geometric verifiers (disjoint code paths) that check containment of all free n-iamonds; optimality follows from the UNSAT certificates.
%D T. R. Dawson, Problem, Fairy Chess Review, Vol. 5, No. 4, 1942.
%H Peter Exley, <a href="https://github.com/pexley-math/oeis-a392363">Solver code, data, and figures</a>, GitHub.
%H The Problemist, <a href="https://www.theproblemist.org/mags.pl?type=fcr&page=volumes">Fairy Chess Review volumes archive</a> (Vol. 5 covers 1942-1945).
%H Puzzle Zapper, <a href="https://puzzlezapper.com/aom/mathrec/polycover.html">Polyomino Common Superforms</a>.
%e For n=4, a(4)=5: contains all 3 free tetraiamonds.
%e For n=5, a(5)=7: contains all 4 free pentaiamonds.
%e For n=11, a(11)=27: first n with a non-contiguous row in the optimal container.
%e For n=14, a(14)=42: contains all 26166 free 14-iamonds.
%Y Cf. A000577 (free n-iamonds), A327094 (square-grid analog), A394840 (smallest polyiamond with n-cell hole), A352029 (number of minimum-size square containers).
%K nonn,hard,more
%O 1,2
%A _Peter Exley_, Apr 06 2026