A175881 Number of closed Knight's tours on a 6 X n board.

0, 0, 0, 0, 8, 9862, 1067638, 55488142, 3374967940, 239187240144, 15360134570696, 964730606632516, 61989683445413228, 4005716717182224826, 255967892553030600920, 16378998506224697063588, 1050504687249683771795632, 67351449674771471216148786, 4314151246752166099728445868

Offset: 1

Author

Johan de Ruiter, Dec 05 2010

Keywords

• nonn

Comments

Could you please say how you calculated these numbers? - N. J. A. Sloane, Dec 05 2010?

I kept track of pairs of loose ends within the two rightmost columns of a 6 X n board, assuming that everything to the left of these two columns is fully connected and that there are no cycles (or one if this is a final state). Next I added a new column and connected it to the rightmost two columns in all ways such that there are no cycles formed(or one if this results in a final state) and the leftmost column in the current state is fully connected and can be dropped. From this followed a transition matrix. I can provide a reference to my writeup once it is completed and has been accepted by my supervisor. - Johan de Ruiter, Dec 05 2010

Examples

			The smallest 6 X n board admitting a closed Knight's tour is the 6 X 5, on which there are 8 such tours.
		

Links

• Johan de Ruiter, Table of n, a(n) for n = 1..40
• J. de Ruiter, Counting Domino Coverings and Chessboard Cycles, 2010.

Crossrefs

A070030 deals with 3 X 2n boards, A175855 with 5 X 2n boards.

Cf. A383662.

Formula

a(n) = A383662(3n). - Don Knuth, May 05 2025