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

0, 1, 0, 1, 1, 4, 7, 31, 95, 420, 1682, 7544, 33288, 152022, 696096, 3231001

Offset: 1

Author

Arthur O'Dwyer, Jan 11 2023

Keywords

• nonn
• more

Comments

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.

This sequence counts ouroboroi which do not designate a specific head or tail cell; thus the unique 8-cell ouroboros is

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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:

##H #HT

# T # #

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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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# # # ## # # # ##

# ## ## # # ## # #

# # #### ## # # #

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Links

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

Crossrefs

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.

A359707 counts free (2-sided) ouroboros polyominoes with k=2n cells. A359706 subtracted from A359707 gives the count of chiral pairs.