cp's OEIS Frontend

A359706 Number of free (2-sided) ouroboros polyominoes with k=2n cells.

%I A359706 #16 Jan 18 2023 09:36:15
%S A359706 0,1,0,1,1,4,7,31,95,420,1682,7544,33288,152022,696096,3231001
%N A359706 Number of free (2-sided) ouroboros polyominoes with k=2n cells.
%C A359706 A "snake" polyomino is a polyomino in which exactly two cells have exactly one (Von Neumann) neighbor apiece, and the rest have two neighbors apiece. Arthur O'Dwyer coined the term "ouroboros polyomino" for a polyomino in which every cell has exactly two neighbors: that is, an ouroboros polyomino is like a "snake" in which the head cell neighbors the tail cell.
%C A359706 A324407 etc. use the term "polyomino ring" in place of "ouroboros polyomino."
%C A359706 A checkerboard coloring shows that every ouroboros must have an even number of cells.
%C A359706 This sequence counts ouroboroi which do not designate a specific head or tail cell; thus the unique 8-cell ouroboros is
%C A359706     ###
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%C A359706 One could imagine counting "headed" ouroboroi, in which the head and tail are distinguished. There are two distinct ways to create a free 8-cell "headed" ouroboros:
%C A359706     ##H  #HT
%C A359706     # T  # #
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%C A359706 This sequence first differs from A359707 (the count of 1-sided ouroboroi) at k=14. The four chiral 14-cell ouroboroi, each of which is counted once by A359706 and twice by A359707, are
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%C A359706     # ##  ##  #  # ##  #  #
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%H A359706 Arthur O'Dwyer, <a href="https://quuxplusone.github.io/blog/2022/12/08/polyomino-snakes/">Polyomino strips, snakes, and ouroboroi</a> (gives the first 32 terms)
%H A359706 Arthur O'Dwyer, <a href="https://quuxplusone.github.io/blog/code/2022-12-08-polyomino-snakes-and-strips.cpp">C++ program</a>
%o A359706 (C++) // see Links section
%Y A359706 A002013 counts free (2-sided) snake polyominoes with k=n cells. A359706 added to A002013 gives the number of free polyominoes in which each cell has at most 2 (Von Neumann) neighbors.
%Y A359706 A359707 counts free (2-sided) ouroboros polyominoes with k=2n cells. A359706 subtracted from A359707 gives the count of chiral pairs.
%K A359706 nonn,more
%O A359706 1,6
%A A359706 _Arthur O'Dwyer_, Jan 11 2023