cp's OEIS Frontend

It is assumed that all edges have length one. - N. J. A. Sloane, Apr 17 2019

These are referred to as 'polysticks', 'polyedges' or 'polyforms'. - Jack W Grahl, Jul 24 2018

Number of connected subgraphs of the square lattice (or grid) containing n length-one line segments. Configurations differing only a rotation or reflection are not counted as different. The question may also be stated in terms of placing unit toothpicks in a connected arrangement on the square lattice. - N. J. A. Sloane, Apr 17 2019

The solution for n=5 features in the card game Digit. - Paweł Rafał Bieliński, Apr 17 2019

With A001411 as main input and counting the symmetrical shapes separately, higher terms can be computed efficiently (see formula). - Bert Dobbelaere, Jan 07 2019

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:

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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).