A323562
Number of rooted self-avoiding king's walks on an infinite chessboard trapped after n moves.
Original entry on oeis.org
8, 200, 2446, 21946, 169782, 1205428, 8119338, 52862872, 336465352, 2108185746
Offset: 8
a(8) = 8, because the following 8 walks of 8 moves of a king starting at S with a first move (0,0)->(1,0) visit all neighbors of the trapping location T. The starting point itself is also blocked. There are no such shortest walks with first move (0,0)->(1,1).
.
o <-- o <-- o o o <-- o o --> o --> o o <-- o <-- o
| ^ ^ \ / ^ ^ | | ^
v | | / \ | | v v |
o --> T o o T o o T o o T o
^ ^ \ \ | | / ^
| | \ \ v v / |
S --> o --> o S --> o --> o S --> o o o S --> o
.
S --> o --> o S --> o --> o S --> o o o S --> o
| | / / ^ ^ \ |
v v / / | | \ v
o --> T o o T o o T o o T o
^ | | \ / | | ^ ^ |
| v v / \ v v | | v
o <-- o <-- o o o <-- o o --> o --> o o <-- o <-- o
- _Hugo Pfoertner_, Jul 23 2020
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
A322831
Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.
Original entry on oeis.org
71, 71, 40, 77, 45, 51, 42, 56, 49, 51, 48, 54
Offset: 3
- S. Hemmer, P. C. Hemmer, An average self-avoiding random walk on the square lattice lasts 71 steps, J. Chem. Phys. 81, 584 (1984)
- Hugo Pfoertner, Examples of self-trapping random walks.
- Hugo Pfoertner, Probability density for the number of steps before trapping occurs, 2018.
- Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk.
- Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.
Cf.
A001668,
A001411,
A001334,
A077482,
A306175,
A306177,
A306178,
A306179,
A306180,
A306181,
A306182.
Cf.
A122223,
A122224,
A122226,
A127399,
A127400,
A127401,
A300665,
A323141,
A323560,
A323562,
A323699.
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))
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.
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.
Showing 1-6 of 6 results.
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