A238117
Number of states with reflective symmetry arising in matrix method for enumerating Hamiltonian cycles on 2n X 2n grid.
Original entry on oeis.org
1, 4, 14, 40, 120, 320, 946, 2496, 7418, 19616
Offset: 1
A238118
Number of continuations with reflective symmetry arising in matrix method for enumerating Hamiltonian cycles on 2n X 2n grid.
Original entry on oeis.org
1, 6, 20, 101, 327, 1560, 5333, 24727, 88422, 403552
Offset: 1
A333864
Number of Hamiltonian cycles on an n X 2*n grid.
Original entry on oeis.org
1, 4, 236, 18684, 32463802, 54756073582, 2365714170297014, 87106950271042689032, 88514516642574170326003422, 71598455565101470929617326988084, 1673219200189416324422979402201514800461, 29815394539834813572600735261571894552950941626, 15836807024750749574106724392556189684881848226515147589
Offset: 2
- Huaide Cheng, Table of n, a(n) for n = 2..16
- Olga Bodroža-Pantić, B. Pantić, I. Pantić AND M. Bodroža-Solarov: Enumeration of Hamiltonian cycles in some grid grafs. MATCH Commun. Math. Comput. Chem. 70:1 (2013), 181-204. on Research Gate.
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# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333864(n):
universe = tl.grid(n - 1, 2 * n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles(is_hamilton=True)
return cycles.len()
print([A333864(n) for n in range(2, 8)])
a(10) and a(12) quoted from Olga's paper.
A181584
Number of cycles of length (2n+1)^2-1 on 2n+1 X 2n+1 square grid.
Original entry on oeis.org
5, 226, 255088, 6663430912, 3916162476483538, 51249820944023435573470, 14870957102232406137455708164254, 95494789899510664733921727510895952184006
Offset: 1
A193346
Number of (directed) Hamiltonian paths on the n X n X n grid graph.
Original entry on oeis.org
1, 144, 4960608, 55493434415544000
Offset: 1
For n = 1, there is a trivial Hamiltonian path of length 0.
For n = 2, the 144 paths fall in three different equivalence classes. Two of the three classes can be derived by taking a Hamiltonian cycle on a cube and deleting a single edge. The third class is a spiral path that ends at the opposite corner from its starting point.
- Raoul D. Schram and Helmut Schiessel, Exact enumeration of Hamiltonian walks on the 4x4x4 cube and applications to protein folding, Journal of Physics A: Mathematical and Theoretical, vol 46 (2013), 485001.
- Raoul D. Schram and Helmut Schiessel, Corrigendum: Exact enumeration of Hamiltonian walks on the 4x4x4 cube and applications to protein folding, Journal of Physics A: Mathematical and Theoretical, vol 49 (2016), 369501.
- Jamie Shepard, Solvability and Difficulty of the Snake Puzzle in the Cube and its Topological Variants, Honors Thesis, Andrews Univ. (2024) Art. No. 290.
- Eric Weisstein's World of Mathematics, Grid Graph
- Eric Weisstein's World of Mathematics, Hamiltonian Path
- Index entries for sequences related to graphs, Hamiltonian
A222201
Write n=3i+j, 0<=j<3; a(n) = number of Hamiltonian cycles on square grid of points of size 2i+2 X 2i+2 (if j=0), 2i+2 X 2i+3 (j=1) or 2i+3 X 2i+4 (j=2).
Original entry on oeis.org
1, 1, 2, 6, 14, 154, 1072, 5320, 301384, 4638576, 49483138, 13916993782, 467260456608, 10754797724124, 14746957510647992, 1076226888605605706, 53540340738182687296, 354282765498796010420944, 56126499620491437281263608, 6040964455632840415885507728, 191678405883294971709423926242394, 65882516522625836326159786165530572
Offset: 0
A238819
Number of Hamiltonian cycles on 4n+2 X 4n+2 grid with at least 90-degree rotational symmetry.
Original entry on oeis.org
1, 2, 204, 510718, 31008619522, 43911490791183200, 1424639466911800364132674, 1048492580133908850434091619741314
Offset: 0
A301648
Number of longest cycles in the n X n grid graph.
Original entry on oeis.org
0, 1, 5, 6, 226, 1072, 255088, 4638576, 6663430912, 467260456608, 3916162476483538, 1076226888605605706, 51249820944023435573470, 56126499620491437281263608, 14870957102232406137455708164254, 65882516522625836326159786165530572, 95494789899510664733921727510895952184006
Offset: 1
Cf.
A137932 (circumference of the (n-1) X (n-1) grid graph).
Cf.
A003763 (number of Hamiltonian cycles in the 2n X 2n grid graph).
Cf.
A181584 (number of longest cycles in the (2n+1) X (2n+1) grid graph).
A238816
Number of Hamiltonian cycles on 2n X 2n grid with at least the symmetry of reflection in an axis.
Original entry on oeis.org
1, 4, 44, 2828, 564468, 754425400, 3079904455096, 88444819222239178, 7685637690960745082050, 4793315937811919497089287562
Offset: 1
A238817
Number of Hamiltonian cycles on 2n X 2n grid with at least 180-degree rotational symmetry.
Original entry on oeis.org
1, 2, 28, 1504, 520176, 696179102, 5373177281748, 166903679914150336, 28636599794306124116062, 20262965974179958448766775754
Offset: 1
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