A333233 Number of free Tangles of size n.

1, 1, 2, 5, 16, 55, 221, 947, 4239, 19452, 90791, 428839, 2043548, 9807941

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

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o-o-o

but this is:

o-o-o

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o-o-o

Links

Table of n, a(n) for n=0..13.

Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2020. Sums of rows from Table 4.1 (C).

Crossrefs

Cf. A019988, A056841, A333080.

Sequence in context: A026106 A308027 A066642 * A019988 A137732 A119611

Adjacent sequences: A333230 A333231 A333232 * A333234 A333235 A333236

Keyword

nonn,hard,more

Author

Douglas A. Torrance, Mar 12 2020

Extensions

a(11)-a(13) from John Mason, Feb 14 2023

Status

approved