OFFSET
0,4
COMMENTS
We say that a free polyomino is prime if it cannot be tiled by any other free polyomino besides the 1 X 1 square and itself.
The tiling of P must be with a single polyomino, and that single polyomino may not be the unique monomino or P itself. For example, decomposing the T-tetromino into a 3 X 1 and a 1 X 1 would use multiple tiles, and this is not permitted.
It can be shown that a(n) > 0 for all n >= 4, by considering the polyomino whose cells are at (0,1), (-1,1), (0,2), and (x,0) for all x = 0, 1, ..., n-4.
LINKS
Cibulis, Liu, and Wainwright, Polyomino Number Theory (I), Crux Mathematicorum, 28(3) (2002), 147-150.
FORMULA
a(n) = A000105(n) if n is prime.
EXAMPLE
For n = 4, the T-tetromino cannot be decomposed into smaller congruent polyominoes:
+---+
| |
+---+ +---+
| |
+-----------+
The other four free tetrominoes can, however:
+---+
| |
| | +---+
| | | |
+---+ | | +---+---+ +---+---+
| | | | | | | | |
| | +---+---+ | | | +---+---+---+
| | | | | | | | |
+---+ +-------+ +---+---+ +---+---+
Thus a(4) = 1.
CROSSREFS
KEYWORD
nonn,hard,more,nice
AUTHOR
Drake Thomas, Mar 11 2021
EXTENSIONS
a(14)-a(17) from John Mason, Sep 16 2022
a(1) corrected by John Mason, Feb 27 2023
STATUS
approved