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%I A348095 #28 Jun 02 2023 15:21:04
%S A348095 0,0,0,1,1,10,42,254,1331,7358,39543,212911,1135876,6039878,31975124,
%T A348095 168790048,888664299,4669177072
%N A348095 Number of free n-polysticks embedded in the square lattice with at least one cycle.
%C A348095 The number of holes h = e - v + 1 (e = the number of edges and v the number of vertices). - _John Mason_, Feb 12 2023
%H A348095 John Mason, <a href="/A348095/a348095.jpg">Examples of size 6</a>
%F A348095 a(n) + A056841(n+1) = A019988(n).
%e A348095 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.
%e A348095 Size 6 examples 1 through 5:
%e A348095   +-+        +-+    +        +    +-+-+
%e A348095   | |        | |    |        |      | |
%e A348095   +-+-+-+  +-+-+-+  +-+-+  +-+-+    +-+-+
%e A348095                     | |    | |
%e A348095                     +-+    +-+
%e A348095 Examples 6 through 10:
%e A348095   +        +-+ +    +      +-+-+  +-+-+
%e A348095   |        | | |    |      | |    |   |
%e A348095   +-+      +-+-+    +-+-+  +-+-+  +-+-+
%e A348095   | |                 | |
%e A348095   +-+-+               +-+
%Y A348095 Cf. A056841 (tree-like), A019988 (free polysticks).
%K A348095 nonn,more
%O A348095 1,6
%A A348095 _R. J. Mathar_, Sep 30 2021
%E A348095 a(14)-a(18) from _John Mason_, Jun 01 2023

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