A237429 -id:A237429 - cp's OEIS frontend

A227257 Number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

0, 1, 24, 1760, 411861, 551247139, 2883245852086, 85948329517780776, 11001968794030973784902, 7462399462450938863305238264

Offset: 1

Christopher Hunt Gribble, Jul 05 2013

			When n = 2, there is only 1 Hamiltonian circuit in a 4 X 4 square lattice, where the orbits under the symmetry group of the square have 4 elements.  The 4 elements are:
    o__o__o__o    o__o__o__o    o__o__o__o    o__o  o__o
    |        |    |        |    |        |    |  |  |  |
    o  o__o__o    o  o__o  o    o__o__o  o    o  o  o  o
    |  |          |  |  |  |          |  |    |  |  |  |
    o  o__o__o    o  o  o  o    o__o__o  o    o  o__o  o
    |        |    |  |  |  |    |        |    |        |
    o__o__o__o    o__o  o__o    o__o__o__o    o__o__o__o

Giovanni Resta, Simple C program for computing a(1)-a(4)
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 2014.

Cf. A003763, A209077, A063524, A227005, A227301.

A063524 + A227005 + A227257 + A227301 = A209077.
1*A063524 + 2*A227005 + 4*A227257 + 8*A227301 = A003763.
a(n) = A237429(n) + A237430(n). - Ed Wynn, Feb 07 2014

a(4) from Giovanni Resta, Jul 11 2013

a(5)-a(10) from Ed Wynn, Feb 05 2014

A237430 Number of nonisomorphic Hamiltonian cycles on 2n X 2n square grid of points with two-fold rotational symmetry (and no other symmetry).

Original entry on oeis.org

0, 0, 5, 366, 129871, 174041330, 1343294003351, 41725919954578785, 7159149948562719664049, 5065741493544986113047994120

Offset: 1

Ed Wynn, Feb 07 2014

			An example of each isomorphism class for n=3.
  o-o-o-o-o-o   o-o-o-o-o-o   o-o-o-o-o-o   o-o-o-o o-o   o-o-o-o o-o
  |         |   |         |   |         |   |     | | |   |     | | |
  o-o-o-o o-o   o o-o-o-o-o   o o-o o-o-o   o-o o-o o o   o o-o o-o o
        | |     | |           | | | |         | |   | |   | | |     |
  o-o-o-o o-o   o o-o-o-o-o   o o o o o-o   o-o o-o-o o   o-o o o-o-o
  |         |   |         |   | | | | | |   |         |       | |
  o-o o-o-o-o   o-o-o-o-o o   o-o o o o o   o o-o-o o-o   o-o-o o o-o
    | |                 | |       | | | |   | |   | |     |     | | |
  o-o o-o-o-o   o-o-o-o-o o   o-o-o o-o o   o o o-o o-o   o o-o o-o o
  |         |   |         |   |         |   | | |     |   | | |     |
  o-o-o-o-o-o   o-o-o-o-o-o   o-o-o-o-o-o   o-o o-o-o-o   o-o o-o-o-o

Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 2014.

Cf. A209077, A227257, A237429.

a(n) = A227257(n) - A237429(n).

cp's OEIS Frontend

A227257 Number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 4 elements.

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