A333249 Number of one-sided Tangles of size n.

1, 1, 2, 7, 25, 99, 415, 1849, 8368, 38712, 181111, 856833, 4085025, 19612082

Offset: 0

Author

Douglas A. Torrance, Mar 13 2020

Keywords

• nonn
• more

Comments

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.

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:

o-o-o

| |

o-o-o

but this is:

o-o-o

| | |

o-o-o

Tangles may also be 'fixed' if we do not allow rotations and reflections (A333080) or 'free' if we allow both rotations and reflections (A333233).

Links

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

Crossrefs

Cf. A151537, A333080, A333233.

Extensions

a(11)-a(13) from John Mason, Feb 15 2023