A078797
Sum of square displacements over all self-avoiding n-step walks on a square lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/A046661(n).
Original entry on oeis.org
1, 8, 41, 176, 679, 2452, 8447, 28120, 91147, 289324, 902721, 2777112, 8441319, 25398500, 75744301, 224156984, 658855781, 1924932324, 5593580859, 16175728584, 46572304083, 133556779740, 381611332725, 1086759598120
Offset: 1
Example: a(2)=8 because the A046661(2)=3 different self-avoiding 2-step walks end at (1,-1),(1,1)->d^2=2 and at (2,0)->d^2=4, so a(2) = 2*2 + 1*4 = 8 a(3)=41 because the end-points of the 9 different 3-step walks are: (0,-1),(0,1)->d^2=1, (1,-2),(1,2),(2,-1),(2,-1),(2,1),(2,1)->d^2=5, (3,0)->d^2=9. a(3) = 2*1 + 6*5 + 1*9 = 41 See also "Distribution of end point distance" at first link
A078798
Sum of Manhattan distances over all self-avoiding n-step walks on square lattice. Numerator of mean Manhattan displacement s(n) = a(n)/A046661(n).
Original entry on oeis.org
1, 6, 23, 80, 263, 834, 2569, 7764, 23095, 67910, 197607, 570560, 1635331, 4661026, 13212739, 37296004, 104836893, 293710714, 820132581, 2283926980, 6343214871, 17578257134, 48604029143, 134141458280, 369519394643
Offset: 1
a(3)=23 because 2 of the A046661(3)=9 walks end at Manhattan distance 1: (0,-1),(0,1) and 7 walks end at Manhattan distance 3: (1,-2),(1,2),2*(2,-1),2*(2,1),(3,0); a(3)=2*1+7*3=23 See also "Distribution of end point distance" at first link.
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).
A078717
Number of n-step self-avoiding walks on cubic lattice with first step specified.
Original entry on oeis.org
1, 5, 25, 121, 589, 2821, 13565, 64661, 308981, 1468313, 6989025, 33140457, 157329085, 744818613, 3529191009, 16686979329, 78955042017, 372953947349, 1762672203269, 8319554639789, 39285015083693, 185296997240401, 874331369198569
Offset: 1
A323140
Number of uncrossed king's paths of length n, reduced for symmetry, A272773/8.
Original entry on oeis.org
1, 7, 45, 280, 1712, 10351, 62082, 370142, 2196701, 12988928, 76572159, 450277842, 2642226994, 15476427641, 90508059371
Offset: 1
A002900
Number of n-step walks on square lattice.
Original entry on oeis.org
2, 6, 18, 50, 142, 390, 1086, 2958, 8134, 22050, 60146, 162466, 440750, 1187222, 3208298, 8622666, 23233338, 62329366, 167558310, 448848582, 1204403014, 3222280242, 8633306906, 23073198658, 61740677454, 164856393110, 440658745814
Offset: 1
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A001411 is the main entry for this sequence.
-
(* b = A001411 *) mo = Tuples[{-1, 1}, 2]; b[0] = 1; b[tg_, p_:{{0, 0}}] := b[tg, p] = Block[{e, mv = Complement[Last[p] + #& /@ mo, p]}, If[tg == 1, Length[mv], Sum[b[tg-1, Append[p, e]], {e, mv}]]];
a[n_] := b[n]/2;
Table[an = a[n]; Print[an]; an, {n, 1, 16}] (* Jean-François Alcover, Nov 02 2018, after Giovanni Resta in A001411 *)
A336448
Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.
Original entry on oeis.org
0, 4, 32, 164, 704, 2716, 9808, 33788, 112480, 364588, 1157296, 3610884, 11108448, 33765276, 101594000, 302977204, 896627936, 2635423124, 7699729296, 22374323436, 64702914336, 186289216332, 534227118960, 1526445330900, 4347038392480, 12341626847324, 34940293640400, 98660244502668
Offset: 0
a(1) = 4 as a single step of length 1 can be taken in four ways on the square lattice the sum of square end-to-end displacements is 4*1 = 4.
a(2) = 32. The two 2-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
+
| 2 +---+---+ 4
+---+
.
The first walk can be taken in 8 ways on a square lattice, the latter in 4 ways, thus the total displacement over all 2-step walks is 8*2 + 4*4 = 32.
a(3) = 164. The five 3-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
+
+---+ | +---+ +
| 1 + 5 | 5 | 5 +---+---+---+ 9
+---+ | +---+ +---+---+
+---+
.
The first four walks can be taken in 8 ways on a square lattice, the last in 4 ways, thus the total displacement over all 3-step walks is 8*1 + 8*5 + 8*5 + 8*5 + 4*9 = 164.
A066372
Number of different shapes formed by bending a piece of wire of length n in the plane.
Original entry on oeis.org
1, 1, 2, 3, 5, 8, 15, 23, 43, 71, 128, 209, 379, 650, 1145, 1928, 3422, 5908, 10295, 17530, 30738, 53088, 91971, 157194, 273621, 471865, 814557, 1393822, 2414895, 4157492, 7160018, 12253782, 21163410, 36381025, 62549316, 107029982, 184430758
Offset: 1
Richard D. Plotz (Dick(AT)Plotz.com), Dec 22 2001
Let LRUD denote left, right, up, down. Then for n = 1..4 the solutions are R, RD, RDL, RDR, RDLU, RDLD, RDRD.
For n=5 the 5 shapes are:
__.__. __. .__ |__. .__. __.
|__| |__| |__| | |__| |__.
|__
- Deborah Freedman, dlf(AT)alumni.princeton.edu, personal communication.
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.
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