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A342430
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Number of prime polyominoes with n cells.
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0
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0, 0, 1, 2, 1, 12, 5, 108, 145, 974, 2210, 17073, 31950, 238591, 587036, 3174686, 9236343, 50107909
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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For n = 4, the T-tetromino cannot be decomposed into smaller congruent polyominoes:
+---+
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+---+ +---+
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+-----------+
The other four free tetrominoes can, however:
+---+
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| | +---+
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+---+ | | +---+---+ +---+---+
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| | +---+---+ | | | +---+---+---+
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+---+ +-------+ +---+---+ +---+---+
Thus a(4) = 1.
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CROSSREFS
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KEYWORD
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nonn,hard,more,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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