A384173 Number of Hamiltonian paths from NW to SW corners in an n X n grid reduced for symmetry, i.e., where reflection about the x-axis is not counted as distinct.

1, 1, 1, 5, 43, 897, 44209, 4467927, 1043906917, 506673590576, 555799435739334, 1284472450789974196, 6625529679919810063544, 72597408139909172033687226, 1762085630816152820582838187465, 91326629994353561722347679614188407

Offset: 1

Author

Oliver R. Bellwood, May 21 2025

Keywords

• nonn
• walk

Comments

When n is odd there are no symmetric Hamiltonian paths from NW to SW corners, and therefore a(n) = A000532(n)/2.

Examples

			The two paths of A000532(3) = 2 are equivalent under reflection about the x-axis:
  + - + - +
          |
  + - +   +
  |   |   |
  +   + - +
  +   + - +
  |   |   |
  + - +   +
          |
  + - + - +
		

References

• J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A: Math. Theor. 40 (2007) 14667-14678.
• J.-M. Mayer, C. Guez and J. Dayantis, Exact computer enumeration of the number of Hamiltonian paths in small square plane lattices, Physical Review B, Vol. 42 Number 1, 1990.

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.

Crossrefs

Cf. A265914, A209077.

Formula

a(n) = A000532(n)/2 for odd n.