A348095 Number of free n-polysticks embedded in the square lattice with at least one cycle.

0, 0, 0, 1, 1, 10, 42, 254, 1331, 7358, 39543, 212911, 1135876, 6039878, 31975124, 168790048, 888664299, 4669177072

Offset: 1

Author

R. J. Mathar, Sep 30 2021

Keywords

• nonn
• more

Comments

The number of holes h = e - v + 1 (e = the number of edges and v the number of vertices). - John Mason, Feb 12 2023

Examples

			The 4-stick with a cycle is the unit square. The 5-stick with a cycle is the unit square with one protruding edge. The 10 6-sticks with a cycle are the unit square with two protruding edges (in various cis, meta, trans configurations) or with a protruding 2-stick at various angles, or a 2x1 rectangle.
Size 6 examples 1 through 5:
  +-+        +-+    +        +    +-+-+
  | |        | |    |        |      | |
  +-+-+-+  +-+-+-+  +-+-+  +-+-+    +-+-+
                    | |    | |
                    +-+    +-+
Examples 6 through 10:
  +        +-+ +    +      +-+-+  +-+-+
  |        | | |    |      | |    |   |
  +-+      +-+-+    +-+-+  +-+-+  +-+-+
  | |                 | |
  +-+-+               +-+
		

Links

• John Mason, Examples of size 6

Crossrefs

Cf. A056841 (tree-like), A019988 (free polysticks).

Formula

a(n) + A056841(n+1) = A019988(n).

Extensions

a(14)-a(18) from John Mason, Jun 01 2023