A385120 Number of fixed tree-like polyedges on the square lattice with n edges, rooted at a vertex.

1, 4, 18, 88, 435, 2184, 11018, 55888, 284229, 1448800, 7396290, 37804344, 193405121, 990117104, 5072380140

Offset: 0

Author

Ben Samberg, Jun 18 2025

Keywords

• nonn
• more
• hard

Examples

			a(0) = 1: empty structure.
a(1) = 4: a single vertical or horizontal edge, rooted at one of the two vertices.
a(2) = 18: six unrooted two-edge polyedges (a straight path oriented in 2 possible ways and an L-shaped path oriented in 4 possible ways), each rooted at one of the three vertices.
		

Links

• Ben Samberg, a(2) visualization
• Ben Samberg, Python algorithm

Crossrefs

Cf. A096267 (not necessarily treelike), A056841 (free), A066158 (polyominoes).

Cf. A308409.

Formula

a(n) = A308409(n) * (n+1). - Andrei Zabolotskii, Jul 02 2025