A359706 -id:A359706 - cp's OEIS frontend

A359707 Number of 1-sided ouroboros polyominoes with k=2n cells.

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

Offset: 1

Arthur O'Dwyer, Jan 11 2023

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.
A324407 etc. use the term "polyomino ring" in place of "ouroboros polyomino."
A checkerboard coloring shows that every ouroboros must have an even number of cells.

Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi (gives the first 32 terms)
Arthur O'Dwyer, C++ program

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.
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
    ###   ####   ###   ###
    # #   #  ##  # #   # ##
    # ##  ##  #  # ##  #  #
    #  #   ####  ## #  #  #
    ####          ###  ####
and their mirror-reflections.

cp's OEIS Frontend

A359707 Number of 1-sided ouroboros polyominoes with k=2n cells.

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