A336281 Total number of ways of embedding connected graphs with n edges in the square lattice with diagonals allowed.

2, 6, 41, 318, 3108, 32243, 350575, 3896568

Offset: 1

Author

James W. Anderson, Jul 15 2020

Keywords

• nonn
• hard
• more
• nice

Comments

The embedding must map edges in the graph onto either horizontal or vertical grid lines of length 1 or diagonals of length sqrt(2). Vertices in the graph must map onto lattice points, and of course must preserve the incidence structure of the graph. A square in the lattice may have both diagonals present - their intersection does not count as an incidence.

Configurations differing only a rotation or reflection are not counted as different.

The resulting figures are variously called 'polysticks', 'polyedges' or 'polyforms'.

Links

• N. J. A. Sloane, Illustration for a(1)=2, a(2)=6, a(3)=41. [Thanks to Peter Munn for correcting errors in my first drawing.]

Crossrefs

Without diagonal edges, we get A019988.

Cf. A052436.

Extensions

a(7)-a(8) from John Mason, Aug 17 2021