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A375770 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions distinct under reflections and rotations.

%I A375770 #18 Sep 04 2024 08:47:44
%S A375770 1,1,10,149,3177,76258,1991098,56431302,1738662461,58282168670,
%T A375770 2121623710614,83566630166058,3545346228604588,161250925229195536,
%U A375770 7827463597195165900,403872784815626357788,22069190323151660044413,1273007854598883147607470,77288239799225577008977654
%N A375770 In an n X n grid draw straight walls between cells, starting at a border, such that the resulting figure is connected and has only one-cell wide paths; a(n) is the number of solutions distinct under reflections and rotations.
%C A375770 This sequence contains some, but not all of the spanning trees in A349718.
%H A375770 Andrew Howroyd, <a href="/A375770/b375770.txt">Table of n, a(n) for n = 1..100</a>
%H A375770 Andrew Howroyd, <a href="/A375770/a375770.gp.txt">PARI Program and formula</a>, Sep 2024.
%e A375770 a(2)=1:
%e A375770 +=======+
%e A375770 | o - o |
%e A375770 | |   | |
%e A375770 | o ║ o |
%e A375770 +===+===+
%e A375770 a(3)=10:
%e A375770 +===========+  +=======+===+  +=======+===+  +===+===+===+  +===========+
%e A375770 | o - o - o |  | o - o ║ o |  | o - o ║ o |  | o ║ o ║ o |  | o - o - o |
%e A375770 | |   |   | |  | |   |   | |  | |   | ║ | |  | |   |   | |  | |   |   ══+
%e A375770 | o ║ o ║ o |  | o ║ o - o |  | o ║ o ║ o |  | o - o - o |  | o ║ o - o |
%e A375770 | | ║ | ║ | |  | | ║ |   | |  | | ║ |   | |  | |   |   | |  | | ║ |   | |
%e A375770 | o ║ o ║ o |  | o ║ o ║ o |  | o ║ o - o |  | o ║ o ║ o |  | o ║ o ║ o |
%e A375770 +===+===+===+  +===+===+===+  +===+=======+  +===+===+===+  +===+===+===+
%e A375770 +===+=======+  +=======+===+  +===========+  +===========+  +=======+===+
%e A375770 | o ║ o - o |  | o - o ║ o |  | o - o - o |  | o - o - o |  | o - o ║ o |
%e A375770 | |   |   ══+  | |   |   | |  | |   |   ══+  +═══  |   ══+  +═══  |   | |
%e A375770 | o - o - o |  | o ║ o - o |  | o ║ o - o |  | o - o - o |  | o - o - o |
%e A375770 | |   |   | |  | | ║ |   ══+  | | ║ |   ══+  | |   |   | |  | |   |   ══+
%e A375770 | o ║ o ║ o |  | o ║ o - o |  | o ║ o - o |  | o ║ o ║ o |  | o ║ o - o |
%e A375770 +===+===+===+  +===+=======+  +===+=======+  +===+===+===+  +===+=======+
%e A375770 n=4 sample
%e A375770 +===+===+===+===+  +=======+===+===+
%e A375770 | o ║ o ║ o ║ o |  | o - o ║ o ║ o |
%e A375770 | |   |   |   | |  +═══  | ║ |   | |
%e A375770 | o - o - o - o |  | o - o ║ o - o |
%e A375770 +═══  |   |   ══+  | |   | ║ |   ══+
%e A375770 | o - o ║ o - o |  | o ║ o ║ o - o |
%e A375770 | |   | ║ |   ══+  | | ║ |   |   | |
%e A375770 | o ║ o ║ o - o |  | o ║ o - o ║ o |
%e A375770 +===+===+=======+  +===+=======+===+
%e A375770 n=5 sample
%e A375770 +===+===+===+===+===+
%e A375770 | o ║ o ║ o ║ o ║ o |
%e A375770 | |   | ║ | ║ |   | |
%e A375770 | o - o ║ o ║ o - o |
%e A375770 | |   |   |   |   ══+
%e A375770 | o ║ o - o - o - o |
%e A375770 | | ║ |   |   |   ══+
%e A375770 | o ║ o ║ o ║ o - o |
%e A375770 | | ║ | ║ | ║ |   | |
%e A375770 | o ║ o ║ o ║ o ║ o |
%e A375770 +===+===+===+===+===+
%e A375770 n=6 sample
%e A375770 +===========+===+===+===+
%e A375770 | o - o - o ║ o ║ o ║ o |
%e A375770 | |   |   | ║ | ║ | ║ | |
%e A375770 | o ║ o ║ o ║ o ║ o ║ o |
%e A375770 | | ║ | ║ |   |   | ║ | |
%e A375770 | o ║ o ║ o - o - o ║ o |
%e A375770 | | ║ | ║ |   |   | ║ | |
%e A375770 | o ║ o ║ o ║ o ║ o ║ o |
%e A375770 | | ║ | ║ | ║ | ║ |   | |
%e A375770 | o ║ o ║ o ║ o ║ o - o |
%e A375770 | | ║ | ║ | ║ | ║ |   ══+
%e A375770 | o ║ o ║ o ║ o ║ o - o |
%e A375770 +===+===+===+===+=======+
%e A375770 Examples of spanning trees where some of the walls do not start at a border, so they are not included in this sequence.
%e A375770 +===+===+=======+  +===============+
%e A375770 | o ║ o ║ o - o |  | o - o - o - o |
%e A375770 | | ║ |   |   | |  +══════════   | |
%e A375770 | o ║ o - o ║ o |  | o - o - o ║ o |
%e A375770 | | ║ ═════ ║ | |  | |   ══  | ║ | |
%e A375770 | o ║ o - o ║ o |  | o ║ o - o ║ o |
%e A375770 | |   |   | ║ | |  | |   ═════   | |
%e A375770 | o - o ║ o - o |  | o - o - o - o |
%e A375770 +=======+=======+  +===============+
%o A375770 (PARI) \\ See Link section for program file.
%o A375770 vector(20, n, A375770(n)) \\ _Andrew Howroyd_, Sep 03 2024
%Y A375770 Cf. A349718, A375817 (not reduced for symmetries), A375859 (up to rotations), A375860 (up to symmetries of rectangle).
%K A375770 nonn
%O A375770 1,3
%A A375770 _Lars Blomberg_, Aug 27 2024
%E A375770 a(1) set to 1 and a(9) onwards from _Andrew Howroyd_, Aug 31 2024