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.
A010575
Number of n-step self-avoiding walks on 4-d cubic lattice.
Original entry on oeis.org
1, 8, 56, 392, 2696, 18584, 127160, 871256, 5946200, 40613816, 276750536, 1886784200, 12843449288, 87456597656, 594876193016, 4047352264616, 27514497698984, 187083712725224, 1271271096363128, 8639846411760440, 58689235680164600, 398715967140863864
Offset: 0
- Hugo Pfoertner, Table of n, a(n) for n = 0..24 [from the Clisby et al. link below]
- 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 A6 for n <= 24.
- Nathan Clisby, Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps, arXiv:1703.10557 [cond-mat.stat-mech], 30 Mar 2017.
- M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
- D. MacDonald, D. L. Hunter, K. Kelly, and N. Jan, Self-avoiding walks in two to five dimensions: exact enumerations and series study, J Phys A: Math Gen 25 (1992) Vol. 6, 1429-1440 [Gives 18 terms]
- A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
- Hugo Pfoertner, Results for the 4D Self-Trapping Random Walk
- Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant
a(18) onwards from
R. J. Mathar using data from Clisby et al, Aug 31 2007
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.
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