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%I #19 Feb 15 2023 13:45:49
%S 1,1,2,7,25,99,415,1849,8368,38712,181111,856833,4085025,19612082
%N Number of one-sided Tangles of size n.
%C a(n) is the number of one-sided 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 'one-sided', we mean that we allow rotations but not reflections.
%C Dual graphs of Tangles are polyedges (A151537), but the only chordless cycles allowed are squares, e.g., this is *not* the dual graph of a Tangle:
%C o-o-o
%C | |
%C o-o-o
%C but this is:
%C o-o-o
%C | | |
%C o-o-o
%C Tangles may also be 'fixed' if we do not allow rotations and reflections (A333080) or 'free' if we allow both rotations and reflections (A333233).
%H Douglas A. Torrance, <a href="https://arxiv.org/abs/1906.01541">Enumeration of planar Tangles</a>, arXiv:1906.01541 [math.CO], 2020. Sums of rows from Table 4.1 (B).
%Y Cf. A151537, A333080, A333233.
%K nonn,more
%O 0,3
%A _Douglas A. Torrance_, Mar 13 2020
%E a(11)-a(13) from _John Mason_, Feb 15 2023