cp's OEIS Frontend

A342430 Number of prime polyominoes with n cells.

%I A342430 #41 Feb 27 2023 11:31:33
%S A342430 0,0,1,2,1,12,5,108,145,974,2210,17073,31950,238591,587036,3174686,
%T A342430 9236343,50107909
%N A342430 Number of prime polyominoes with n cells.
%C A342430 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.
%C A342430 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.
%C A342430 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.
%H A342430 Cibulis, Liu, and Wainwright, <a href="http://www.paulsalomon.com/uploads/2/8/3/3/28331113/polyomino_number_theory_(i).pdf">Polyomino Number Theory (I)</a>, Crux Mathematicorum, 28(3) (2002), 147-150.
%F A342430 a(n) = A000105(n) if n is prime.
%e A342430 For n = 4, the T-tetromino cannot be decomposed into smaller congruent polyominoes:
%e A342430       +---+
%e A342430       |   |
%e A342430   +---+   +---+
%e A342430   |           |
%e A342430   +-----------+
%e A342430 The other four free tetrominoes can, however:
%e A342430   +---+
%e A342430   |   |
%e A342430   |   |    +---+
%e A342430   |   |    |   |
%e A342430   +---+    |   |         +---+---+        +---+---+
%e A342430   |   |    |   |         |   |   |        |       |
%e A342430   |   |    +---+---+     |   |   |    +---+---+---+
%e A342430   |   |    |       |     |   |   |    |       |
%e A342430   +---+    +-------+     +---+---+    +---+---+
%e A342430 Thus a(4) = 1.
%Y A342430 Cf. A000105, A125759, A213376.
%K A342430 nonn,hard,more,nice
%O A342430 0,4
%A A342430 _Drake Thomas_, Mar 11 2021
%E A342430 a(14)-a(17) from _John Mason_, Sep 16 2022
%E A342430 a(1) corrected by _John Mason_, Feb 27 2023