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
An example of each isomorphism class for n=3.
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A237431
Number of nonisomorphic Hamiltonian cycles on 2n X 2n square grid of points with exactly two axes of reflective symmetry.
Original entry on oeis.org
0, 1, 3, 20, 244, 6891, 378813, 47917598, 12118420172, 6998287399637
Offset: 1
Examples of 2 of the 3 classes for n=3. Note that all examples also have two-fold (but not four-fold) rotational symmetry.
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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
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.
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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.
Original entry on oeis.org
1, 1, 1, 5, 43, 897, 44209, 4467927, 1043906917, 506673590576, 555799435739334, 1284472450789974196, 6625529679919810063544, 72597408139909172033687226, 1762085630816152820582838187465, 91326629994353561722347679614188407
Offset: 1
The two paths of A000532(3) = 2 are equivalent under reflection about the x-axis:
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- 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.
A358212
a(n) is the maximal possible sum of squares of the side lengths of an n^2-gon supported on a subset 1 <= x,y <= n of an integer lattice.
Original entry on oeis.org
4, 10, 36, 98, 232
Offset: 2
- Oliver Mantas Ališauskas, Grid connector, Web application for this problem.
- Oliver Mantas Ališauskas, Giedrius Alkauskas, and Valdas Dičiūnas, Full Grid Lattice Polygons with Maximal Sum of Squares of Edge-Lengths, arXiv:2311.03011 [math.CO], 2023-2024.
- S. Chow, A. Gafni, and P. Gafni, Connecting the dots: maximal polygons on a square grid, Math. Mag. 94 (2021), no. 2, 118-124.
- G. L. Cohen and E. Tonkes, Dartboard arrangements, Elect. J. Combin., 8(2) (2001), #R4.
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