cp's OEIS Frontend

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

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but this is:

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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).

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

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o-o-o

but this is:

o-o-o

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o-o-o

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

| | |

o-o-o

a(n-1) is the number of bond trees with n sites (vertices) on the square lattice.

a(n) is the number of fixed polyedges (A096267) with n edges and no cycles.