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

%I #16 Feb 01 2022 00:30:51

%S 0,0,0,4,12,28,136,488,2624,12420,64120,313584,1675648,8536072

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

%C A quasi-magic tour is a special type of semi-magic tour in which the non-magic lines add to only two different values. Such tours are of interest on boards where magic tours don't exist.

%H G. P. Jelliss, <a href="https://www.mayhematics.com/j/gpj26.htm">Quasi-Magic Tours</a>, The Games and Puzzles Journal #26, April-June 2003.

%H Awani Kumar, <a href="https://arxiv.org/abs/1802.09340">Studies in Tours of Knight on Rectangular Boards</a>, arXiv:1802.09340 [math.GM], 2018.

%e Example of 4 X 14 quasi-magic knight's tour. All short lines sum to 114 and the long lines sum to 397 and 401, that is, 399 +- 2.

%e 3 56 25 32 5 52 21 36 9 48 11 40 15 44

%e 26 29 4 53 24 33 8 49 20 37 18 43 12 41

%e 55 2 31 28 51 6 35 22 47 10 39 14 45 16

%e 30 27 54 1 34 23 50 7 38 19 46 17 42 13

%Y Cf. A328816, A329370.

%K nonn,hard,more

%O 1,4

%A _Awani Kumar_, Nov 14 2019