OFFSET
0,3
COMMENTS
a(n) is the number of free Tangles (smooth simple closed curves piecewise-defined by quadrants of circles) which have a dual graph containing n edges, or equivalently, enclose an area of (4*n + Pi)*r^2, where 1/r is the curvature. By 'free', we mean that we allow rotations and reflections.
Tangles may also be 'fixed', i.e., if we do not allow rotations and reflections (A333080).
Tangles whose dual graphs are trees correspond exactly to diagonal polyominoes (A056841).
Dual graphs of Tangles are polysticks (A019988), but the only chordless cycles allowed are squares, e.g., this is *not* the dual graph of a Tangle:
o-o-o
| |
o-o-o
but this is:
o-o-o
| | |
o-o-o
LINKS
Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2020. Sums of rows from Table 4.1 (C).
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Douglas A. Torrance, Mar 12 2020
EXTENSIONS
a(11)-a(13) from John Mason, Feb 14 2023
STATUS
approved