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A003763 Number of Hamiltonian cycles on 2n X 2n square grid of points. 21
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.

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

A. Poenitz [Po"nitz], 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: Math. Gen. 17 (1984) 445-453.


Artem M. Karavaev, Table of n, a(n) for n=1..11 [From Artem M. Karavaev, Sep 29 2010]

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.

Artem M. Karavaev, Hamilton Cycles. [From Artem M. Karavaev, Sep 29 2010]

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



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Last modified April 23 17:34 EDT 2014. Contains 240946 sequences.