A333233 Number of free Tangles of size n.

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

Offset: 0

Author

Douglas A. Torrance, Mar 12 2020

Keywords

• nonn
• hard
• more

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

Cf. A019988, A056841, A333080.

Extensions

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