A237431 -id:A237431 - cp's OEIS frontend

A227005 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 2 elements.

0, 1, 4, 20, 346, 6891, 634172, 47917598, 27622729933, 6998287399637

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 2 elements.  The 2 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

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, A227257, A227301.

A063524 + A227005 + A227257 + A227301 = A209077.
1*A063524 + 2*A227005 + 4*A227257 + 8*A227301 = A003763.
a(2n) = A237431(2n), a(2n+1) = A237431(2n+1) + A237432(n+1). -  Ed Wynn, Feb 07 2014

a(4) from Giovanni Resta, Jul 11 2013

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

A237432 Number of nonisomorphic Hamiltonian cycles on (4n-2) X (4n-2) square grid of points with four-fold rotational symmetry (and no other symmetry).

Original entry on oeis.org

0, 1, 102, 255359, 15504309761, 21955745395591600, 712319733455900182066337, 524246290066954425217045809870657

Offset: 1

Ed Wynn, Feb 07 2014

For square grids of m X m points, there are solutions only for m = (4n-2).

			The two cycles counted as a single class for n=2. These are isomorphic (here meaning isomorphic under the full symmetry group of the square), since each is a reflection of the other.
  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, A227005, A237431.

a(n) = A238819(n-1) / 2 for n > 1. - Andrew Howroyd, Apr 06 2016

a(6)-a(8) from Andrew Howroyd, Apr 06 2016

cp's OEIS Frontend

A227005 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 2 elements.

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