OFFSET
1,1
COMMENTS
From Peter Exley, Jun 04 2026: (Start)
A fixed n-iamond is a connected set of n unit triangles, counted up to translation only (rotations and reflections distinct). Their number is A001420(n), with A001420(1) = 2.
a(n) is the minimum number of cells in a connected polyiamond containing every fixed n-iamond as a translated subset. The first differences are non-monotone, no low-degree polynomial fits, and a(n) exceeds the free-piece analog A392363(n). The reported a(n) is the minimum container size over the n X n triangular search window (1 X 2 for n=1); it is the conjectured global minimum, proved minimal within that window. (End)
LINKS
Peter Exley, Paper and figures, GitHub.
EXAMPLE
For n = 1, a(1) = 2: the 2-cell rhombus formed by one up-pointing and one down-pointing unit triangle contains both fixed 1-iamonds.
For n = 2, a(2) = 4: contains all 3 fixed 2-iamonds.
For n = 3, a(3) = 6: contains all 6 fixed 3-iamonds; the 6-cell hexagon is an optimal container.
For n = 9, a(9) = 31: contains all 1838 fixed 9-iamonds, within a 6 X 9 bounding box; the optimum has a 2-row 9-cell bulk, a 2-row 5-cell middle band, with two 1-cell caps above the bulk and one 1-cell cap below the band, saving 2 cells over the row-bounded construction.
CROSSREFS
Cf. A001420 (number of fixed n-iamonds), A000577 (number of free n-iamonds), A006534 (number of one-sided n-iamonds), A392363 (smallest polyiamond containing all free n-iamonds), A327094 (square-grid analog, free pieces), A000217 (triangular numbers; conjecturally coincides with the hex-grid fixed-piece container sequence for all n, proved for n = 1..7).
KEYWORD
nonn,hard,more
AUTHOR
Peter Exley, Apr 22 2026
EXTENSIONS
a(10)-a(11) from Peter Exley, Jun 04 2026
STATUS
approved