A328931
Number of Hamiltonian paths in an n X n square, starting from an edge, finishing anywhere, all symmetries excluded.
Original entry on oeis.org
1, 1, 4, 51, 660, 30745, 1621471, 312637285, 72599875346, 60968508324409, 64128000370443037, 240651566540823214362, 1162174738476331286327484, 19776621796151182708398884540, 441809773825445785471324877668710
Offset: 1
All distinct paths through a 1 X 1 labyrinth visiting all cells.
+ +
|**|
+--+
.
All distinct paths through a 2 X 2 labyrinth visiting all cells.
+ +--+
| |**|
+ + +
| |
+--+--+
.
All distinct paths through a 3 X 3 labyrinth visiting all cells.
+ +--+--+
| | |
+ + + +
| | |
+--+--+ +
|** |
+--+--+--+
.
+ +--+--+
| | **|
+ + +--+
| | |
+ +--+ +
| |
+--+--+--+
.
+ +--+--+
| | |
+ + + +
| |**| |
+ +--+ +
| |
+--+--+--+
.
+ +--+--+
| | |
+ + + +
| | | |
+ + + +
| |**|
+--+--+--+
A341269
Number of non-extendable self-avoiding walks in an n X n grid starting at the top left corner.
Original entry on oeis.org
1, 2, 20, 548, 40440, 8442742, 5088482972, 8963926817126, 46591697187961736
Offset: 1
Example of a self-avoiding walk on a 3 X 3 grid that visits every node (Hamiltonian path):
.
1---2---3
|
6---5---4
|
7---8---9
.
Two examples of a self-avoiding walk on a 3 X 3 grid that do not visit every node:
.
1---2 7
| |
X 3 6
| |
X 4---5
.
or
.
1 8---7
| |
2---3 6
| |
X 4---5
.
A363577
Number of inequivalent Hamiltonian paths starting in the lower left corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent).
Original entry on oeis.org
1, 1, 3, 23, 347, 10199, 683227, 85612967, 25777385143, 14396323278040, 19799561204761862, 50351228336401026361, 319210377672595552740369, 3736517399241599771428011100, 109790442395888863208285555153329, 5952238893391106787883489313797219949
Offset: 1
There are 3 paths for n=3:
+--+--+ +--+--+ +--+ +
| | | | | | |
+ + + + +--+ + + +
| | | | | | | |
+ +--+ + +--+ + +--+
A fourth path:
+--+--+
|
+--+ +
| | |
+ +--+
is the same as the second one in the row above after a 90-degree rotation.
All paths starting E are the same as the corresponding ones starting N after reflection in the forward diagonal.
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