A333080 Number of fixed Tangles of size n.

1, 2, 6, 22, 88, 372, 1626, 7292, 33309, 154374, 723740, 3425124, 16336747, 78437858

Offset: 0

Author

Douglas A. Torrance, Mar 07 2020

Keywords

• nonn
• hard
• more

Comments

a(n) is the number of fixed 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 'fixed', we mean that we do not allow rotations or reflections.

Dual graphs of Tangles are polyedges (A096267), 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

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o-o-o

Links

• Douglas A. Torrance, Enumeration of planar Tangles, arXiv:1906.01541 [math.CO], 2019-2020. Sums of rows from Table 4.1 (A).

Crossrefs

Dual graphs of Tangles which are trees are bond trees on the square lattice (A308409), free Tangles (A333233).

Extensions

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