A359707 - OEIS

%I #12 Jan 18 2023 09:36:27

%S 0,1,0,1,1,4,11,45,178,762,3309,14725,66323,302342,1391008,6453950

%N Number of 1-sided ouroboros polyominoes with k=2n cells.

%C 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 A324407 etc. use the term "polyomino ring" in place of "ouroboros polyomino."

%C A checkerboard coloring shows that every ouroboros must have an even number of cells.

%H 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 Arthur O'Dwyer, <a href="https://quuxplusone.github.io/blog/code/2022-12-08-polyomino-snakes-and-strips.cpp">C++ program</a>

%o (C++) // see Links section

%Y A151514 counts 1-sided snake polyominoes with k=n cells. A359707 added to A151514 gives the number of 1-sided polyominoes in which each cell has at most 2 (Von Neumann) neighbors.

%Y A359706 counts free (2-sided) ouroboros polyominoes with k=2n cells. A359707 minus A359706 gives the count of chiral pairs. This sequence first differs from A359706 at k=14; the four chiral pairs of 14-cell ouroboroi are

%Y     ###   ####   ###   ###

%Y     # #   #  ##  # #   # ##

%Y     # ##  ##  #  # ##  #  #

%Y     #  #   ####  ## #  #  #

%Y     ####          ###  ####

%Y and their mirror-reflections.

%K nonn,more

%O 1,6

%A _Arthur O'Dwyer_, Jan 11 2023