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.
A078605
Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).
Original entry on oeis.org
1, 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212, 64177763220925, 322314275563424, 1613192327878789, 8049191357609204, 40048773875769449, 198750753713937600
Offset: 1
a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.
A323857
Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on 4-d cubic lattice.
Original entry on oeis.org
1, 14, 135, 1144, 9083, 69690, 522781, 3864524, 28243251, 204687550, 1473038447, 10542725976, 75096139471, 532846305962, 3767808141891, 26566180648012, 186826646453453
Offset: 1
a(3) = 135, because there are 6 (of A010575(3)/8=49) end points with Manhattan distance 1, (0,-1,0,0), (0,1,0,0), (0,0,-1,0), (0,0,1,0), (0,0,0,-1), (0,0,0,1), and the remaining 43 end points all have Manhattan distance 3, e.g., (3,0,0,0), (2,-1,0,0), ..., (1,-1,-1,0), ... 135 = 6*1 + 43*3.
A079158
Sum of end-to-end Manhattan distances over all self-avoiding walks on cubic lattice trapped after n steps.
Original entry on oeis.org
5, 40, 399, 2472, 17436, 98400, 601626, 3238694, 18355742, 96020478
Offset: 11
a(12)=40 because the A077817(12)=20 trapped walks stop at 5*(1,1,0)->d=2, 5*(2,0,0)->d=2, 10*(1,0,1)->d=2, so a(12)=5*2+5*2+10*2=40. See "Enumeration of all self-trapping walks of length 12" at link.
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