

A342430


Number of prime polyominoes with n cells.


0



0, 1, 1, 2, 1, 12, 5, 108, 145, 974, 2210, 17073, 31950, 238591
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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 Ttetromino 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, ..., n4.


LINKS

Table of n, a(n) for n=0..13.
Cibulis, Liu, and Wainwright, Polyomino Number Theory (I), Crux Mathematicorum, 28(3) (2002), 147150.


FORMULA

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


EXAMPLE

For n = 4, the Ttetromino 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



