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A342430 Number of prime polyominoes with n cells. 0

%I

%S 0,1,1,2,1,12,5,108,145,974,2210,17073,31950,238591

%N Number of prime polyominoes with n cells.

%C We say that a free polyomino is prime if it cannot be tiled by any other free polyomino besides the 1x1 square and itself.

%C 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.

%C 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 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 a(n) = A000105(n) if n is prime.

%e For n = 4, the T-tetromino cannot be decomposed into smaller congruent polyominoes:

%e +---+

%e | |

%e +---+ +---+

%e | |

%e +-----------+

%e The other four free tetrominoes can, however:

%e +---+

%e | |

%e | | +---+

%e | | | |

%e +---+ | | +---+---+ +---+---+

%e | | | | | | | | |

%e | | +---+---+ | | | +---+---+---+

%e | | | | | | | | |

%e +---+ +-------+ +---+---+ +---+---+

%e Thus a(4) = 1.

%Y Cf. A000105, A125759, A213376.

%K nonn,hard,more,nice

%O 0,4

%A _Drake Thomas_, Mar 11 2021

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Last modified September 19 16:34 EDT 2021. Contains 347564 sequences. (Running on oeis4.)