A333080 Number of fixed Tangles of size n.

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

Offset

0,2

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

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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], 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).

Sequence in context: A049137 A287223 A365246 * A096267 A150264 A294593

Adjacent sequences: A333077 A333078 A333079 * A333081 A333082 A333083

Keyword

nonn,hard,more

Author

Douglas A. Torrance, Mar 07 2020

Extensions

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

Status

approved