A323559
Number of rooted self-avoiding knight's paths of length n on an infinite chessboard with first move specified.
Original entry on oeis.org
1, 7, 49, 337, 2323, 15805, 107737, 727619, 4921655, 33056939, 222323989, 1487064391, 9957971965, 66391431607, 443085643919, 2946553003837, 19611967535129, 130149475953673
Offset: 1
A323560
Number of self-avoiding knight's paths trapped after n moves on an infinite chessboard with first move specified.
Original entry on oeis.org
1728, 10368, 332660, 1952452
Offset: 15
There are two (of a(15)=1728) paths of 15 moves of minimum extension 5 X 5:
(N) b1 d2 e4 c5 a4 b2 d1 e3 d5 b4 a2 c1 e2 d4 b5 c3, and
(N) a4 c5 e4 d2 b1 a3 b5 d4 e2 c1 a2 b4 d5 e3 d1 c3.
A376736
a(n) is the numerator of the expected number of random moves of a chess knight to reach a position outside an nXn chessboard, starting in one of the corners.
Original entry on oeis.org
1, 1, 4, 62, 269, 1766, 395497, 101338, 44125237, 227721959, 3361699348115, 483866477194862, 277887411827604127, 790848403160840410, 2785714552717079970073201, 89715505143567836216964174, 2034961072108249587083318018747, 457177774768288408431166142758841, 1085703228381446052419019696184520372520
Offset: 1
1, 1, 4/3, 62/43, 269/167, 1766/1017, 395497/213488, 101338/51901, 44125237/21578387, 227721959/106983448, ...
Approximately 1, 1, 1.333, 1.442, 1.611, 1.736, 1.853, 1.953, 2.045, 2.129, 2.206, ...
A376737 are the corresponding denominators.
-
\\ Uses function droprob from A376606
knightmoves = [[2, 1], [1, 2], [-1, 2], [-2, 1], [-2, -1], [-1, -2], [1, -2], [2, -1]];
a376736(n) = numerator(droprob(n, knightmoves, 8))
A323132
Number of uncrossed unrooted knight's paths of length n on an infinite board.
Original entry on oeis.org
1, 6, 25, 160, 966, 6018, 37079, 227357
Offset: 1
See illustrations at Pfoertner link.
A323561
Number of rooted self-avoiding king's walks of n moves on an infinite chessboard with first move specified.
Original entry on oeis.org
2, 14, 92, 584, 3644, 22482, 137626, 837466, 5072590, 30611376, 184171252, 1105262004, 6618842522, 39564403462, 236123357538, 1407249202976, 8376673823516
Offset: 1
A323133
Number of symmetric uncrossed unrooted knight's paths of length n on an infinite board.
Original entry on oeis.org
1, 6, 7, 29, 46, 170, 299, 969
Offset: 1
A323699
Number of uncrossed knight's walks as specified in A323700, counting isomorphisms only once.
Original entry on oeis.org
1, 8, 56, 404, 2563, 16516, 102280, 639532, 3899662
Offset: 4
In algebraic chess notation, the two walks double counted in A323700(7) are
N c4 d2 e4 c5 a4 b2 d1 c3 and N d4 c2 e3 d5 b4 a2 c1 b3.
A356404
The number of closed routes of the chess knight, different in shape, consisting of 2 * n jumps on a checkered field without repeating cells of the route.
Original entry on oeis.org
1, 3, 25, 480, 11997, 350275, 10780478
Offset: 1
a(1)=1, since the only closed route for n=1 can be considered two jumps of a chess knight from any cell and back;
a(2)=3 because the chess knight has 3 closed routes consisting of 4 jumps. We list them in the coordinate plane Oxy:
1st route: (0;0), (1;2), (3;3), (2;1), (0;0);
2nd route: (0;1), (1;3), (3;2), (2;0), (0;1);
3rd route: (0;1), (2;2), (4;1), (2;0), (0;1).
A323700
Number of rooted uncrossed knight's walks on an infinite chessboard trapped after n moves with first move specified.
Original entry on oeis.org
1, 8, 56, 406, 2572, 16596, 102654, 642441, 3914084
Offset: 4
a(4) = 1 because there is only one trapped walk of 4 moves, written in algebraic chess notation: (N) b1 d2 b3 a1 c2.
For longer walks see link to illustrations in A323699.
A072174
Maximum path length of a crippled knight on an n X n board.
Original entry on oeis.org
1, 1, 5, 9, 16, 27, 38, 51, 66
Offset: 1
For 3 X 3, the longest path is:
1 . 3
4 . .
. 2 5
The knight cannot move from #5 because it would have to cross over 2 or 3, so a(3)=5.
For 8 X 8, a(8)=51 has a unique solution:
. 1 8 19 22 25 28 31
7 20 23 26 29 32 . .
2 9 18 21 24 27 30 33
. 6 3 10 17 34 37 40
4 11 16 35 38 41 . .
49 46 5 12 15 36 39 42
. . 50 47 44 13 . .
51 48 45 14 . . 43 .
Best known solution for 9 X 9 (66 moves):
. 56 53 50 47 44 27 . .
. . . 55 52 49 46 43 28
57 54 51 48 45 42 29 26 .
64 61 58 41 38 35 32 . 30
. . 65 62 59 40 37 34 25
66 63 60 39 36 33 24 31 .
. 2 5 8 11 14 17 20 23
4 7 10 13 16 19 22 . .
1 . 3 6 9 12 15 18 21
- A crippled knight is defined by Dario Uri in the Journal of Recreational Mathematics, problem 2465, Vol. 29 #4.
- Vol. 30 #4 has an example for 8 X 8 with 48 moves found by Henry Ibstedt.
Showing 1-10 of 10 results.
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