OFFSET
1,3
COMMENTS
Equivalently, a(n) is the least number of free rooted (or pointed) polyiamonds (A369365) corresponding to a given polyiamond with n cells.
From Peter Exley, Apr 07 2026: (Start)
a(19) = 5 and a(20) = 6 were computed by exhaustive enumeration of all free polyiamonds combined with Burnside orbit counting under the dihedral group D_6. Two independent verifiers with disjoint orbit-counting code confirmed all values for n = 1..20.
The maximum possible symmetry group order for a polyiamond depends on n mod 6. For n = 6k+5, no rotational symmetry is possible (n = 2 mod 3 blocks 3-fold rotation; n odd blocks 180-degree rotation), giving |G_max| = 2 and a(6k+5) >= ceiling(n/2) = 3(k+1). For n = 6k+2, 180-degree rotation is available but 3-fold rotation is not, giving |G_max| = 4 (the Klein 4-group) and a(6k+2) >= ceiling(n/4).
The extremal 19-iamonds (achieving a(19) = 5) have D_3 symmetry (|G| = 6), while the extremal 20-iamonds (achieving a(20) = 6) have Klein 4-group symmetry (|G| = 4). The number of minimizers (consistent with the companion sequence A369367): 2 for n = 19, 17 for n = 20. Our minimizer counts were verified against all 18 known A369367 terms.
a(19) = 5 and a(20) = 6 break the constant-block pattern seen for k = 1, 2: at those blocks a(6k+1) through a(6k+4) were all equal (3,3,3,3 and 4,4,4,4), but at k = 3 the first two positions diverge.
Conjecture: a(6k+5) = 3(k+1) for all k >= 0. Observed for k = 0, 1, 2 (n = 5, 11, 17). The next test case is a(23), predicted to be 12.
(End)
LINKS
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Pontus von Brömssen, Jan 22 2024
EXTENSIONS
a(18) from John Mason, Sep 20 2024
a(19)-a(20) from Peter Exley, Apr 07 2026
STATUS
approved