A396483 Number of undirected Hamiltonian cycles in the mod n square grid graph, reduced for symmetry (with rotations, reflections, and translations not counted as distinct).

1, 3, 2, 19, 138, 11591, 657781

Offset

1,2

Comments

Here a "mod n square grid graph" is a n X n graph with wrapping allowed, e.g., a path can make a 1 unit step between the left and the right edge (same row) or the top and the bottom edge (same column) of the graph.

Links

Table of n, a(n) for n=1..7.

Example

a(4) = 19: 10 symmetric cases:
  :                :                :                :
  +--+--+--+       +--+--+--+       +--+--+--+       +--+--+--+       +--+--+--+
           |                |                |                |       |        |
..+--+--+  +..   ..+--+--+  +..     +--+  +--+       +--+--+--+       +  +--+  +
        |                |          |  |  |          |                |  |  |  |
..+--+  +--+..   ..+--+--+  +..   ..+  +  +  +..     +--+--+--+       +--+  +  +
     |                      |          |  |  |                |             |  |
..+  +--+--+..     +--+--+--+       +--+  +--+       +--+--+--+     ..+--+--+  +..
  :                :                :                :
                                                           :                :  :
  +--+--+--+       +--+--+--+       +--+--+--+       +--+  +--+       +--+  +--+
  |        |       |        |       |        |       |  |     |       |  |
  +  +--+  +       +--+  +--+       +--+  +--+     ..+  +--+  +..   ..+  +--+  +..
  |  |  |  |          |  |             |  |                |                |  |
  +  +  +  +       +--+  +--+     ..+  +  +  +..     +--+  +--+     ..+  +--+  +..
  |  |  |  |       |        |       |  |  |  |       |  |     |       |  |
  +--+  +--+       +--+--+--+       +--+  +--+     ..+  +--+  +..   ..+  +--+  +..
                                                           :                :  :
plus 9 nonsymmetric cases:
  :                :                :                :
  +--+--+--+       +--+--+--+       +--+--+--+       +--+--+--+       +--+--+--+
           |                |                |                |       |        |
..+  +--+  +..   ..+  +--+  +..     +--+--+  +       +--+--+  +       +--+--+  +
  |  |  |          |  |  |          |     |  |       |     |  |             |  |
  +--+  +--+     ..+  +  +  +..   ..+  +--+  +..     +--+  +--+     ..+--+  +  +..
           |          |  |  |          |                |                |  |
  +--+--+--+       +--+  +--+     ..+  +--+--+..   ..+  +--+--+..   ..+--+  +--+..
  :                :                :                :
                                             :                :
  +--+--+--+       +--+--+--+       +--+--+  +       +--+--+  +
  |        |       |        |       |     |  |       |     |  |
  +--+--+  +       +--+--+  +       +--+  +--+       +--+  +--+
        |  |             |  |          |                |
..+  +--+  +..     +--+  +  +     ..+  +  +--+..   ..+  +  +--+..
  |  |             |  |  |  |       |  |  |          |  |  |
..+  +--+--+..   ..+  +--+  +..   ..+  +--+  +..     +--+  +--+
                                             :                :

Crossrefs

Cf. A222199, A396482 (Hamiltonian paths), A209077 (wrapping disallowed).

Sequence in context: A154262 A367548 A154261 * A098655 A065038 A123225

Adjacent sequences: A396480 A396481 A396482 * A396484 A396485 A396486

Keyword

nonn,walk,hard,more

Author

Charles L. Hohn, May 27 2026

Status

approved