A383660
Number of closed knight's tours in the first 2n cells of a 3 X ceiling(2n/3) board.
Original entry on oeis.org
4, 0, 4, 24, 16, 56, 306, 176, 456, 2632, 1536, 4828, 26788, 15424, 44952, 254288, 147728, 448032, 2502568, 1448416, 4310048, 24228704, 14060048, 42195584, 236335248, 136947616, 409403328, 2297294496, 1332257856, 3989883552, 22366625344, 12965578752, 38798663104, 217604833360, 126169362176
Offset: 11
For n=11 the a(11)=4 solutions are
1 4 7 10 17 20 15 12
6 9 2 21 14 11 18
3 22 5 8 19 16 13 ;
1 4 7 14 11 20 9 18
6 15 2 21 8 17 12
3 22 5 16 13 10 19 ;
1 4 21 12 15 6 17 8
20 11 2 5 18 9 14
3 22 19 10 13 16 7 ;
1 4 21 18 9 6 11 14
20 17 2 5 12 15 8
3 22 19 16 7 10 13 .
- Donald E. Knuth, Hamiltonian paths and cycles, Prefascicle 8a of The Art of Computer Programming (work in progress, 2025).
- Don Knuth, Table of n, a(n) for n = 11..150
- Don Knuth, CWEB program with input parameter board,100,3,0,0,5,0,0.gb [the graph "board(50, 6, 0, 0, 5, 0, 0)" generated by the Stanford GraphBase.
A383661
Number of closed knight's tours in the first 2n cells of a 5 X ceiling(2n/5) board.
Original entry on oeis.org
1, 0, 1, 30, 0, 148, 8, 78, 9309, 612, 62749, 44202, 42049, 2916485, 147192, 18284136, 13311268, 13008389, 973107552, 51147756, 6190192748, 4557702762, 4311375354, 316985255470, 16552301184, 2015267424300, 1495135512514, 1417634375316, 104324890543686, 5459334927260, 663068761241948
Offset: 9
For n=9 the a(9)=1 example is
1 14 5 10
4 9 2 15
13 18 11 6
8 3 16
17 12 7 .
- Donald E. Knuth, Hamiltonian paths and cycles. Prefascicle 8a of The Art of Computer Programming (work in progress, 2025).
- Don Knuth, Table of n, a(n) for n = 9..150
- Don Knuth, CWEB program with input parameter board,60,5,0,0,5,0,0.gb [the graph "board(50, 6, 0, 0, 5, 0, 0)" generated by the Stanford GraphBase].
A383663
Number of closed knight's tours in the first 2n cells of a 7 X ceiling(2n/7) board.
Original entry on oeis.org
2, 11, 58, 0, 21, 1020, 9309, 1481, 34162, 1295034, 1067638, 2213327, 50139185, 682189688, 144994543, 2607067351, 53099426601, 34524432316, 57716933870, 1388556345255, 16330667126220, 3697750041989, 70341043737487, 1662805965511580, 1250063279938854, 2122662114673944
Offset: 11
For n=11, the first of a(11)=2 solutions is
1 4 21 6
20 7 2
3 22 5
8 19 10
11 16 13
14 9 18
17 12 15
and the other is obtained by reflecting the bottom four rows:
1 4 21 6
20 7 2
3 22 5
10 19 8
13 16 11
18 9 14
15 12 17 .
- Donald E. Knuth, Hamiltonian paths and cycles. Prefascicle 8a of The Art of Computer Programming (work in progress, 2025).
- Don Knuth, Table of n, a(n) for n = 11..147
- Don Knuth, CWEB program with input parameter board,42,7,0,0,5,0,0.gb [the graph "board(50, 6, 0, 0, 5, 0, 0)" generated by the Stanford GraphBase].
A383664
Number of closed knight's tours in the first 2n cells of an 8 X ceiling(2n/8) board.
Original entry on oeis.org
4, 12, 212, 0, 50, 4525, 101730, 44202, 66034, 2408624, 69362264, 55488142, 101343548, 2398536889, 43391615822, 34524432316, 52661182514, 1231713564493, 20780788492646, 13267364410532, 21515340977481, 552407941427835, 10211663162678661, 7112881119092574, 11873618786859165
Offset: 13
For n=13 the a(13)=4 solutions are
1 4 25 12
24 11 2 5
3 26 13
10 23 6
7 14 9
22 17 20
19 8 15
16 21 18 ;
1 4 25 12
24 11 2 5
3 26 13
10 23 6
7 14 9
20 15 22
15 8 19
18 21 16 ;
1 14 25 22
24 21 2 15
13 26 23
20 3 16
17 12 19
4 9 6
7 18 11
10 5 8 ;
1 14 25 22
24 21 2 15
13 26 23
20 3 16
17 12 19
6 9 4
11 18 7
8 5 10 .
- Donald E. Knuth, Hamiltonian paths and cycles, Prefascicle 8a of The Art of Computer Programming (work in progress, 2025).
- Don Knuth, Table of n, a(n) for n = 13..96
- Don Knuth, CWEB program with input parameter board,32,8,0,0,5,0,0.gb [the graph "board(50, 6, 0, 0, 5, 0, 0)" generated by the Stanford GraphBase].
A175881
Number of closed Knight's tours on a 6 X n board.
Original entry on oeis.org
0, 0, 0, 0, 8, 9862, 1067638, 55488142, 3374967940, 239187240144, 15360134570696, 964730606632516, 61989683445413228, 4005716717182224826, 255967892553030600920, 16378998506224697063588, 1050504687249683771795632, 67351449674771471216148786, 4314151246752166099728445868
Offset: 1
The smallest 6 X n board admitting a closed Knight's tour is the 6 X 5, on which there are 8 such tours.
Showing 1-5 of 5 results.
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