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 A359706 Number of free (2-sided) ouroboros polyominoes with k=2n cells. 2
 0, 1, 0, 1, 1, 4, 7, 31, 95, 420, 1682, 7544, 33288, 152022, 696096, 3231001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 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 ### # # ### 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 # # ### ### 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 ### #### ### ### # # # ## # # # ## # ## ## # # ## # # # # #### ## # # # #### ### #### LINKS Table of n, a(n) for n=1..16. Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi (gives the first 32 terms) Arthur O'Dwyer, C++ program PROG (C++) // see Links section 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. Sequence in context: A358586 A102666 A123801 * A156228 A218959 A283332 Adjacent sequences: A359703 A359704 A359705 * A359707 A359708 A359709 KEYWORD nonn,more AUTHOR Arthur O'Dwyer, Jan 11 2023 STATUS approved

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Last modified September 28 23:23 EDT 2023. Contains 365739 sequences. (Running on oeis4.)