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
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 1-sided ouroboros polyominoes with k=2n cells. A359706 subtracted from A359707 gives the count of chiral pairs.
A121769 counts fixed ouroboros polyominoes.
KEYWORD
nonn,more,changed
AUTHOR
Arthur O'Dwyer, Jan 11 2023
STATUS
approved