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).

1, 1, 3, 23, 347, 10199, 683227, 85612967, 25777385143, 14396323278040, 19799561204761862, 50351228336401026361, 319210377672595552740369, 3736517399241599771428011100, 109790442395888863208285555153329, 5952238893391106787883489313797219949

Offset: 1

Author

Lars Blomberg, Jun 10 2023

Keywords

• nonn
• hard
• walk

Comments

Equivalently, number of inequivalent Hamiltonian paths starting in a corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent). - Martin Ehrenstein, Jul 08 2023

Examples

			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.
		

Links

• Oliver R. Bellwood, Heitor P. Casagrande, and William J. Munro, Fractal Path Strategies for Efficient 2D DMRG Simulations, arXiv:2507.11820 [cond-mat.str-el], 2025. See p. 4.
• Eric Weisstein's World of Mathematics, Grid Graph
• Eric Weisstein's World of Mathematics, Hamiltonian Path
• Index entries for sequences related to graphs, Hamiltonian

Crossrefs

Cf. A000532, A001184, A003763, A007764, A120443, A121785, A121789, A145157, A271507, A384173.

Extensions

a(1) added by N. J. A. Sloane, Jun 10 2023

a(8)-a(9) from Martin Ehrenstein, Jul 08 2023

a(10)-a(16) from Oliver R. Bellwood, Jun 06 2025