cp's OEIS Frontend

A328931 Number of Hamiltonian paths in an n X n square, starting from an edge, finishing anywhere, all symmetries excluded.

%I A328931 #26 Mar 12 2024 19:41:33
%S A328931 1,1,4,51,660,30745,1621471,312637285,72599875346,60968508324409,
%T A328931 64128000370443037,240651566540823214362,1162174738476331286327484,
%U A328931 19776621796151182708398884540,441809773825445785471324877668710
%N A328931 Number of Hamiltonian paths in an n X n square, starting from an edge, finishing anywhere, all symmetries excluded.
%C A328931 Given an n X n grid, start from any outside edge, enter the grid, and visit every square. 1 X 1 is a trivial example. 2 X 2 can only be traversed clockwise or counterclockwise (therefore considered the same solution). For 3 X 3 with the cells labeled ABC/DEF/GHI, the four solutions are ADEBCFIHG, ADGHIFEBC, ADGHIFCE and ADGHEBCFI. All others are rotations or reflections.
%C A328931 Discovered programmatically by exhaustive recursive search.
%H A328931 David Lawrence, <a href="/A328931/a328931_1.txt">All paths up to 4 X 4</a>
%e A328931 All distinct paths through a 1 X 1 labyrinth visiting all cells.
%e A328931   +  +
%e A328931   |**|
%e A328931   +--+
%e A328931 .
%e A328931 All distinct paths through a 2 X 2 labyrinth visiting all cells.
%e A328931   +  +--+
%e A328931   |  |**|
%e A328931   +  +  +
%e A328931   |     |
%e A328931   +--+--+
%e A328931 .
%e A328931 All distinct paths through a 3 X 3 labyrinth visiting all cells.
%e A328931   +  +--+--+
%e A328931   |  |     |
%e A328931   +  +  +  +
%e A328931   |     |  |
%e A328931   +--+--+  +
%e A328931   |**      |
%e A328931   +--+--+--+
%e A328931 .
%e A328931   +  +--+--+
%e A328931   |  |   **|
%e A328931   +  +  +--+
%e A328931   |  |     |
%e A328931   +  +--+  +
%e A328931   |        |
%e A328931   +--+--+--+
%e A328931 .
%e A328931   +  +--+--+
%e A328931   |  |     |
%e A328931   +  +  +  +
%e A328931   |  |**|  |
%e A328931   +  +--+  +
%e A328931   |        |
%e A328931   +--+--+--+
%e A328931 .
%e A328931   +  +--+--+
%e A328931   |  |     |
%e A328931   +  +  +  +
%e A328931   |  |  |  |
%e A328931   +  +  +  +
%e A328931   |     |**|
%e A328931   +--+--+--+
%Y A328931 Cf. A120443, A121789, A145157, A265914.
%K A328931 nonn,more
%O A328931 1,3
%A A328931 _David Lawrence_, Oct 31 2019
%E A328931 a(8)-a(15) from _Andrew Howroyd_, Oct 31 2019