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 A342430 Number of prime polyominoes with n cells. 0
 0, 1, 1, 2, 1, 12, 5, 108, 145, 974, 2210, 17073, 31950, 238591 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS We say that a free polyomino is prime if it cannot be tiled by any other free polyomino besides the 1x1 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 3x1 and a 1x1 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 Cf. A000105, A125759, A213376. Sequence in context: A107722 A167128 A332749 * A181417 A048743 A049055 Adjacent sequences:  A342427 A342428 A342429 * A342431 A342432 A342433 KEYWORD nonn,hard,more,nice AUTHOR Drake Thomas, Mar 11 2021 STATUS approved

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Last modified July 30 17:21 EDT 2021. Contains 346359 sequences. (Running on oeis4.)