%I #21 Mar 09 2023 02:15:26

%S 1,1,3,6,25,84,397,1855,9708,51684,286011,1609097,9222409,53543338,

%T 314612803

%N Number of free polyominoes of size 2n for which there exists at least one closed path that passes through each square exactly once.

%C A polyomino for which more than one closed path exists counts as 1. On the other hand, in A266549, distinct closed paths count separately. For example for n=7, this latter sequence distinguishes between

%C +-+ +-+

%C | | | |

%C + +-+ +-+

%C | |

%C +-+-+-+-+

%C and

%C +-+-+-+

%C | |

%C + +-+ +-+

%C | | | |

%C +-+ +-+-+

%H John Mason, <a href="/A361288/a361288.pdf">Examples</a>

%e For n = 4 the a(4) = 3 solutions are:

%e XXX XX XXXX

%e X X XXX XXXX

%e XXX XXX

%Y Cf. A266549 (where distinct closed paths count separately).

%K nonn,more,hard

%O 2,3

%A _John Mason_ and _Tanya Khovanova_, Mar 07 2023

%E a(13) - a(16) from _Bert Dobbelaere_, Mar 09 2023