A118313
Sum of squared end-to-end distances of all n-step self-avoiding walks on the simple cubic lattice.
Original entry on oeis.org
0, 6, 72, 582, 4032, 25566, 153528, 886926, 4983456, 27401502, 148157880, 790096950, 4166321184, 21760624254, 112743796632, 580052260230, 2966294589312, 15087996161382, 76384144381272, 385066579325550, 1933885653380544, 9679153967272734, 48295148145655224, 240292643254616694, 1192504522283625600, 5904015201226909614, 29166829902019914840, 143797743705453990030, 707626784073985438752, 3476154136334368955958, 17048697241184582716248, 83487969681726067169454, 408264709609407519880320, 1993794711631386183977574, 9724709261537887936102872, 47376158929939177384568598, 230547785968352575619933376
Offset: 0
- R. D. Schram, G. T. Barkema, R. H. Bisseling, Table of n, a(n) for n = 0..36
- N. Clisby, R. Liang and G. Slade Self-avoiding walk enumeration via the lace expansion J. Phys. A: Math. Theor. vol. 40 (2007) p 10973-11017, Table A5 for n<=30.
- A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.
- D. MacDonald, S. Joseph, D. L. Hunter, L. L. Mosley, N. Jan and A. J. Guttmann, Self-avoiding walks on the simple cubic lattice,J Phys A: Math Gen 33 (2000) No 34, 5973-5983
- Raoul D. Schram, Gerard T. Barkema, Rob H. Bisseling, Exact enumeration of self-avoiding walks, J Stat. Mech. (2011) P06019.
A079156
Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.
Original entry on oeis.org
10, 67, 396, 2201, 11870, 62571, 324896, 1665349, 8457890, 42605267, 213305636, 1061939193, 5263752278, 25984214383, 127848694424, 627084275649, 3067923454498
Offset: 2
a(2)=10 because the A078717(2)=5 different self-avoiding 2-step walks end at (1,0,-1),(1,0,1),(1,-1,0),(1,1,0),(2,0,0)->d=2. a(2)=5*2=10. See also "Distribution of end point distance" at Pfoertner link
A323856
Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.
Original entry on oeis.org
1, 16, 177, 1696, 14995, 126180, 1025707, 8133544, 63274143, 484966972, 3672258385, 27533213880, 204715798387, 1511417062948, 11090886972237, 80957709527896, 588206815480213, 4256231985648516, 30685328305245631, 220504966309520728, 1579874958814261407
Offset: 1
a(1) = 1 is the square displacement of the fixed initial step.
a(2) = 16, because one of the A010575(2)/8 = 7 end points is (2,0,0,0) with square distance 4 and the other 6 end points (1,-1,0,0), (1,1,0,0), (1,0,-1,0), (1,0,1,0), (1,0,0,-1), (1,0,0,1) all have square distance 2. 16 = 1*4 + 6*2.
a(3) = 177, because there are 6 end points with square distance 1, e.g., (0,1,0,0), 24 end points with square distance 3, e.g., (1,1,1,0), 18 end points with square distance 5, e.g., (2,1,0,0), and 1 end point with square distance 9, (3,0,0,0). 177 = 6*1 + 24*3 + 18*5 + 1*9.
A079157
Sum of square displacements over all self-avoiding walks on cubic lattice trapped after n steps. Numerator of mean square displacement a(n)/A077817(n).
Original entry on oeis.org
5, 50, 529, 3870, 28900, 185014, 1191698, 7080332, 42072344, 238337862
Offset: 11
a(12)=50 because the A077817(12)=20 trapped walks stop at 5*(1,1,0)->d^2=2, 5*(2,0,0)->d^2=4, 10*(1,0,1)->d^2=2. So, a(12)=5*2+5*4+10*2=50. See "Enumeration of all self-trapping walks of length 12" at link.
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