1,2

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.

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.

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.

Peter Tittmann, More results

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.

Sequence in context: A159865 A004806 A125536 this_sequence A113159 A159717 A160857

Adjacent sequences: A003760 A003761 A003762 this_sequence A003764 A003765 A003766

nonn

Jeffrey Shallit (shallit(AT)graceland.uwaterloo.ca), Feb 14 2002

Two more terms from Andre Poenitz [Andre' Po"nitz] 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