A120443 -id:A120443 - cp's OEIS frontend

A003763 Number of (undirected) Hamiltonian cycles on a 2n X 2n square grid of points.

1, 6, 1072, 4638576, 467260456608, 1076226888605605706, 56126499620491437281263608, 65882516522625836326159786165530572, 1733926377888966183927790794055670829347983946, 1020460427390768793543026965678152831571073052662428097106

Offset: 1

Jeffrey Shallit, Feb 14 2002

Orientation of the path is not important; you can start going either clockwise or counterclockwise.
The number is zero for a 2n+1 X 2n+1 grid (but see A222200).
These are also called "closed rook tours".

			a(1) = 1 because there is only one such path visiting all nodes of a square.

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

N. J. A. Sloane, Table of n, a(n) for n = 1..13 (first 11 terms from _Artem M. Karavaev_, Sep 29 2010; a(12) and a(13) from Pettersson, 2014, added by _N. J. A. Sloane_, Jun 05 2015)
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
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.
Artem M. Karavaev, Hamilton Cycles.
Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.
Ville Pettersson, Graph Algorithms for Constructing and Enumerating Cycles and Related Structures, Dissertation, Aalto, Finland, 2015.
A. Pönitz, Computing invariants in graphs of small bandwidth, Mathematics in Computers and Simulation, 49(1999), 179-191.
A. Pönitz, Über eine Methode zur Konstruktion... PhD Thesis (2004) C.3.
T. G. Schmalz, G. E. Hite and D. J. Klein, Compact self-avoiding circuits on two-dimensional lattices, J. Phys. A 17 (1984), 445-453.
N. J. A. Sloane, Illustration of a(2) = 6.
Peter Tittmann, Enumeration in graphs: counting Hamiltonian cycles. [Broken link?]
Peter Tittmann, Enumeration in graphs: counting Hamiltonian cycles. [Backup copy of top page only, on the Internet Archive]
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv preprint arXiv:1402.0545 [math.CO], 2014.
Index entries for sequences related to graphs, Hamiltonian.

Other enumerations of Hamiltonian cycles on a square grid: A120443, A140519, A140521, A222200, A222201.

a(n) = A321172(2n,2n). - Robert FERREOL, Apr 01 2019

Two more terms from Andre Poenitz [André Pönitz] and Peter Tittmann (poenitz(AT)htwm.de), Mar 03 2003
a(8) from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 21 2006
a(9) and a(10) from Jesper L. Jacobsen (jesper.jacobsen(AT)u-psud.fr), Dec 12 2007

A000532 Number of Hamiltonian paths from NW to SW corners in an n X n grid.

Original entry on oeis.org

1, 1, 2, 8, 86, 1770, 88418, 8934966, 2087813834, 1013346943033, 1111598871478668, 2568944901392936854, 13251059359839620127088, 145194816279817259193401518, 3524171261632305641165676374930, 182653259988707123426135593460533473

Offset: 1

Russ Cox, Mar 15 1996

Number of walks reaching each cell exactly once.

Oliver R. Bellwood, Table of n, a(n) for n = 1..20 (first 19 terms from KeyTo9(AT)Fans, Ed Wynn)
KeyTo9(AT)Fans, Counting paths in a grid - Chinese web page giving the sequence up to 18 items.
Douglas M. McKenna, Tendril Motifs for Space-Filling, Half-Domino Curves, in: Bridges Finland Conference Proceedings, 2016, pp. 119-126.
Douglas M. McKenna, Are Maximally Unbalanced Hilbert-Style Square-Filling Curve Motifs a Drawing Medium?, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 91-98.

Main diagonal of A271592.
Cf. A181688, A181689, A014524, A014585.
Cf. A001184, A145157, A120443, A003763, A271507, A007764, A121785, A121789.
See also A350148.

More terms from Zhao Hui Du, Jul 08 2008
Edited by Franklin T. Adams-Watters, Jul 03 2009
Name clarified by Andrew Howroyd, Apr 10 2016

A209077 Number of Hamiltonian circuits (or self-avoiding rook's tours) on a 2n X 2n grid reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

Original entry on oeis.org

1, 2, 149, 580717, 58407763266, 134528361351329451, 7015812452562871283559623, 8235314565328229583744138065519908, 216740797236120772990979350241355889872437894, 127557553423846099192878370713500303677609606263171680998

Offset: 1

N. J. A. Sloane, Mar 04 2012

nonn

Christopher Hunt Gribble confirms a(3), and reports that there are 121 figures with group of order 1, 24 with group of order 2, and 4 with group of order 4. Then 121*(8/1) + 24*(8/2) + 4*(8/4) = 1072 = A003763(3), 121 + 24 + 4 = 149 = a(3). - N. J. A. Sloane, Feb 22 2013

Jon Wild, Posting to Sequence Fans Mailing List, Dec 10 2011.

Mathoverflow, Counting Hamiltonian cycles in n x n square grid, question asked by Joseph O'Rourke, Jul 25 2018.
Jon Wild, Illustration of a(3)
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv:1402.0545 [math.CO], 3 Feb 2014.
Index entries for sequences related to graphs, Hamiltonian

Cf. A003763, A120443, A140519, A140521.

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

A332307 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 14, 20, 14, 1, 1, 22, 62, 62, 22, 1, 1, 32, 132, 276, 132, 32, 1, 1, 44, 336, 1006, 1006, 336, 44, 1, 1, 58, 688, 3610, 4324, 3610, 688, 58, 1, 1, 74, 1578, 12010, 26996, 26996, 12010, 1578, 74, 1, 1, 92, 3190, 38984, 109722, 229348, 109722, 38984, 3190, 92, 1

Offset: 1

Andrew Howroyd, Feb 09 2020

			Array begins:
================================================
m\n | 1  2   3     4      5       6        7
----+-------------------------------------------
  1 | 1  1   1     1      1       1        1 ...
  2 | 1  4   8    14     22      32       44 ...
  3 | 1  8  20    62    132     336      688 ...
  4 | 1 14  62   276   1006    3610    12010 ...
  5 | 1 22 132  1006   4324   26996   109722 ...
  6 | 1 32 336  3610  26996  229348  1620034 ...
  7 | 1 44 688 12010 109722 1620034 13535280 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
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.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path

Rows n=1..9 are A000012, A003682, A003685, A003695, A003778, A145402, A358794, A358795, A358796.
Main diagonal is A120443.
Cf. A064298, A231829, A271465, A271592, A288518, A321172.

T(n,m) = T(m,n).

A145157 Number of Greek-key tours on an n X n board; i.e., self-avoiding walks on n X n grid starting in top left corner.

Original entry on oeis.org

1, 2, 8, 52, 824, 22144, 1510446, 180160012, 54986690944, 29805993260994, 41433610713353366, 103271401574007978038, 660340630211753942588170, 7618229614763015717175450784, 225419381425094248494363948728158

Offset: 1

Nathaniel Johnston, Oct 03 2008

The sequence may be enumerated using standard methods for counting Hamiltonian cycles on a modified graph with two additional nodes, one joined to a corner vertex and the other joined to all other vertices. - Andrew Howroyd, Nov 08 2015

Nathaniel Johnston, On Maximal Self-Avoiding Walks
Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.

Main diagonal of A378938.
Cf. A046994, A046995, A145156, A160240, A160241.
Cf. A000532, A001184, A120443, A003763, A271507, A007764.

a(9)-a(15) from Andrew Howroyd, Nov 08 2015

A288518 Array read by antidiagonals: T(m,n) = number of (undirected) paths in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 3, 12, 3, 6, 49, 49, 6, 10, 146, 322, 146, 10, 15, 373, 1618, 1618, 373, 15, 21, 872, 7119, 14248, 7119, 872, 21, 28, 1929, 28917, 111030, 111030, 28917, 1929, 28, 36, 4118, 111360, 801756, 1530196, 801756, 111360, 4118, 36

Offset: 1

Andrew Howroyd, Jun 10 2017

Paths of length zero are not counted here.

			Table starts:
=================================================================
m\n|  1    2      3       4         5          6            7
---|-------------------------------------------------------------
1  |  0    1      3       6        10         15           21 ...
2  |  1   12     49     146       373        872         1929 ...
3  |  3   49    322    1618      7119      28917       111360 ...
4  |  6  146   1618   14248    111030     801756      5493524 ...
5  | 10  373   7119  111030   1530196   19506257    235936139 ...
6  | 15  872  28917  801756  19506257  436619868   9260866349 ...
7  | 21 1929 111360 5493524 235936139 9260866349 343715004510 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..276
Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Grid Graph

Rows 2-7 are A288516, A288527, A358800, A358801, A358802, A358803.
Main diagonal is A288032.
Cf. A231829, A064298, A271465, A271592, A120443, A001184, A145157, A003763.

A271507 Number of self-avoiding walks of any length from NW to SW corners on an n X n grid or lattice.

Original entry on oeis.org

1, 2, 11, 178, 8590, 1246850, 550254085, 741333619848, 3046540983075504, 38141694646516492843, 1453908228148524205711098, 168707605740228097581729005751, 59588304533380500951726150179910606, 64061403305026776755367065417308840021540

Offset: 1

Andrew Howroyd, Apr 08 2016

nonn

Main diagonal of A271465.

Cf. A007764, A000532, A001184, A145157, A120443, A003763.

A271465 = Cases[Import["https://oeis.org/A271465/b271465.txt", "Table"], {, }][[All, 2]];
a[n_] := A271465[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Sep 23 2019 *)

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A271507(n):
    if n == 1: return 1
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n
    paths = GraphSet.paths(start, goal)
    return paths.len()
print([A271507(n) for n in range(1, 10)])  # Seiichi Manyama, Mar 21 2020

A121789 "Spanning Hamiltonian walks" on the square lattice (see Jensen web site for further information).

Original entry on oeis.org

2, 8, 34, 650, 12014, 1016492, 83761994, 32647369000, 12227920752840, 22181389298814376, 38166266554504010420, 323646210116765453608746, 2574827340090912815899810042, 102299512403818451392332665527950

Offset: 1

N. J. A. Sloane, Aug 30 2006

nonn

Number of Hamiltonian paths in the graph P_{n+1} X P_{n+1} starting at any of the n+1 vertices on one side of the graph and terminating at any of the n+1 vertices on the opposite side. - Andrew Howroyd, Apr 10 2016

I. Jensen, Table of n, a(n) for n = 1..18 [from the Jensen link below]
I. Jensen, Series Expansions for Self-Avoiding Walks
I. Jensen, Spanning Hamiltonian walks

Cf. A120443, A121785, A215527.

A268838 Number of (undirected) Hamiltonian paths in the torus grid graph C_n X C_n.

Original entry on oeis.org

1, 4, 756, 45696, 2955700, 560028096, 126412047692, 93784124187136

Offset: 1

Andrew Howroyd, Feb 14 2016

Here, X (sometimes also written \square) is the graph Cartesian product.

Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A003763, A222199, A120443, A268894.

A143246 Number of (directed) Hamiltonian circuits in the n X n grid graph.

Original entry on oeis.org

0, 2, 0, 12, 0, 2144, 0, 9277152, 0, 934520913216, 0, 2152453777211211412, 0, 112252999240982874562527216, 0, 131765033045251672652319572331061144, 0, 3467852755777932367855581588111341658695967892, 0

Offset: 1

Eric W. Weisstein, Aug 01 2008

nonn

Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv preprint arXiv:1402.0545 [math.CO], 2014.
Index entries for sequences related to graphs, Hamiltonian

Cf. A003763 (number of undirected cycles on 2n X 2n grid graph).
Cf. A222065 (A222065(n) = a(2n)).
Cf. A120443.

cp's OEIS Frontend

A003763 Number of (undirected) Hamiltonian cycles on a 2n X 2n square grid of points.

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A000532 Number of Hamiltonian paths from NW to SW corners in an n X n grid.

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A209077 Number of Hamiltonian circuits (or self-avoiding rook's tours) on a 2n X 2n grid reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A332307 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph.

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A145157 Number of Greek-key tours on an n X n board; i.e., self-avoiding walks on n X n grid starting in top left corner.

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A288518 Array read by antidiagonals: T(m,n) = number of (undirected) paths in the grid graph P_m X P_n.

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A271507 Number of self-avoiding walks of any length from NW to SW corners on an n X n grid or lattice.

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A121789 "Spanning Hamiltonian walks" on the square lattice (see Jensen web site for further information).

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A268838 Number of (undirected) Hamiltonian paths in the torus grid graph C_n X C_n.

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A143246 Number of (directed) Hamiltonian circuits in the n X n grid graph.

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