This site is supported by donations to The OEIS Foundation.



Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 4500 articles have referenced us, often saying "we would not have discovered this result without the OEIS".

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003763 Number of Hamiltonian cycles on 2n X 2n square grid of points. 23
1, 6, 1072, 4638576, 467260456608, 1076226888605605706, 56126499620491437281263608, 65882516522625836326159786165530572, 1733926377888966183927790794055670829347983946, 1020460427390768793543026965678152831571073052662428097106 (list; graph; refs; listen; history; text; internal format)



Orientation of the path is not important; you can start going either clockwise or counter-clockwise.

The number is zero for a 2n+1 X 2n+1 grid (but see A222200).

These are also called "closed rook tours".


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


Artem M. Karavaev and 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.

A. Pönitz, Computing invariants in graphs of small bandwidth, Mathematics in Computers and Simulation, 49(1999), 179-191

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 Hamilton cycles

Ed Wynn, Enumeration of nonisomorphic Hamiltonian cycles on square grid graphs, arXiv preprint arXiv:1402.0545, 2014

Index entries for sequences related to graphs, Hamiltonian


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


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

Sequence in context: A159865 A004806 A125536 * A179853 A190351 A219165

Adjacent sequences:  A003760 A003761 A003762 * A003764 A003765 A003766




Jeffrey Shallit, Feb 14 2002


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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 28 09:50 EST 2015. Contains 264557 sequences.