A077482
Number of self-avoiding walks on square lattice trapped after n steps.
Original entry on oeis.org
1, 2, 11, 25, 95, 228, 752, 1860, 5741, 14477, 42939, 109758, 317147, 818229, 2322512, 6030293, 16900541, 44079555, 122379267, 320227677, 882687730, 2315257359, 6346076015, 16675422679, 45502168379, 119728011251, 325510252108, 857400725204
Offset: 7
a(7) = 1 because there is only 1 self-trapping walk with 7 steps: (0,0)(1,0)(1,1)(1,2)(0,2)(-1,2)(-1,1)(0,1); a(8) = 2 because there are 2 self-trapping walks with 8 steps: (0,0)(1,0)(2,0)(2,1)(2,2)(1,2)(0,2)(0,1)(1,1) and (0,0)(1,0)(1,1)(2,1)(3,1)(3,0)(3,-1)(2,-1)(2,0).
A323131
Number of uncrossed rooted knight's paths of length n on an infinite board.
Original entry on oeis.org
1, 7, 47, 303, 1921, 11963, 74130, 454484, 2779152, 16882278, 102384151, 618136584, 3727827148, 22408576099, 134595908277, 806452390868
Offset: 1
a(1) = 1: The fixed initial move.
a(2) = 7: Relative to the direction given by the initial move, there are 7 possible direction changes. The backward direction is illegal for the self-avoiding uncrossed path. Only for the right angle turn its mirror image would coincide with the turn in the opposite direction. Therefore this move would be eliminated in the unrooted walks, making a(2) > A323132(2) = 6.
a(3) = 47: 2 of all 7*7 = 49 continuation moves already lead to a crossing of the first path segment.
See illustrations at Pfoertner link.
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
Original entry on oeis.org
1, 3, 14, 39, 134, 362, 1114, 2974, 8715, 23192, 66131, 175889, 493036, 1311265, 3633777, 9664070, 26564611, 70644166, 193023433, 513251110, 1395938840, 3711196199, 10057272214, 26732694893, 72234863272, 191962874523, 517473126631, 1374873851835
Offset: 7
a(16) = 1 + 2 + 11 + 25 + 95 + 228 + 752 + 1860 + 5741 + 14477 = 23192.
- B. D. Hughes, Random Walks and Random Environments, Vol. I OUP, 1995.
- N. Madras & G. Slade, The Self-Avoiding Walk, Birkhäuser, 1993.
A381979
Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the square lattice.
Original entry on oeis.org
Cf.
A378903 (The expected walk length on the cubic lattice).
Cf.
A077483 (Probability of the occurrence of each walk length).
Showing 1-5 of 5 results.
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