A333080 - OEIS

%I #26 Feb 19 2023 12:13:26

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

%T 78437858

%N Number of fixed Tangles of size n.

%C 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.

%C 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:

%C     o-o-o

%C     |   |

%C     o-o-o

%C   but this is:

%C     o-o-o

%C     | | |

%C     o-o-o

%H Douglas A. Torrance, <a href="https://arxiv.org/abs/1906.01541">Enumeration of planar Tangles</a>, arXiv:1906.01541 [math.CO], 2019-2020. Sums of rows from Table 4.1 (A).

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

%K nonn,hard,more

%O 0,2

%A _Douglas A. Torrance_, Mar 07 2020

%E a(11)-a(13) from _John Mason_, Feb 14 2023