A330610 Number of near-magic (only short lines are magic) knight's tours on a 4 X 2n board.

0, 0, 0, 4, 16, 72, 244, 1012, 3976, 18440, 81252, 388188, 1800728, 8769964

Offset: 1

Author

Awani Kumar, Dec 20 2019

Keywords

• nonn
• hard
• more

Comments

A semi-magic knight's tour is a tour that adds to a constant sum in every line in one direction, but not in every line in the perpendicular direction. A near-magic tour is a special type of semi-magic tour in which the non-magic direction has the lines add to the magic constant and two other values. Such tours are of interest on boards where magic tours don't exist.

Examples

			Example of 4 X 14 near-magic knight's tour. All short lines add to 114 and the long lines add to the magic constant 399 and 399 +- 4.
   1 54 25 32  5 36 23 50 19 44  9 40 15 46
  28 31  4 53 24 51 20 35  8 39 14 45 10 41
  55  2 29 26 33  6 37 22 49 18 43 12 47 16
  30 27 56  3 52 21 34  7 38 13 48 17 42 11
		

Links

• G. P. Jelliss, Semi-Magic Knight's Tours.
• Awani Kumar, Studies in Tours of Knight on Rectangular Boards, arXiv:1802.09340 [math.GM], 2018.

Crossrefs

Cf. A328816, A329483.