|
%I #40 Nov 03 2023 11:16:29
%S 1,8,56,392,2696,18584,127160,871256,5946200,40613816,276750536,
%T 1886784200,12843449288,87456597656,594876193016,4047352264616,
%U 27514497698984,187083712725224,1271271096363128,8639846411760440,58689235680164600,398715967140863864
%N Number of n-step self-avoiding walks on 4-d cubic lattice.
%C The computation for n=16 took 11.5 days CPU time on a 500MHz Digital Alphastation. The asymptotic behavior lim n->infinity a(n)/mu^n=const is discussed in the MathWorld link. The Pfoertner link provides an illustration of the asymptotic behavior indicating that the connective constant mu is in the range [6.79,6.80]. - _Hugo Pfoertner_, Dec 14 2002
%C Computation of the new term a(17) took 16.5 days CPU time on a 1.5GHz Intel Itanium 2 processor. - _Hugo Pfoertner_, Oct 19 2004
%H Hugo Pfoertner, <a href="/A010575/b010575.txt">Table of n, a(n) for n = 0..24</a> [from the Clisby et al. link below]
%H N. Clisby, R. Liang, and G. Slade, <a href="https://doi.org/10.1088/1751-8113/40/36/003">Self-avoiding walk enumeration via the lace expansion</a>, J. Phys. A: Math. Theor., vol. 40 (2007), p. 10973-11017, Table A6 for n <= 24.
%H Nathan Clisby, <a href="https://arxiv.org/abs/1703.10557">Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps</a>, arXiv:1703.10557 [cond-mat.stat-mech], 30 Mar 2017.
%H M. E. Fisher and D. S. Gaunt, <a href="http://dx.doi.org/10.1103/PhysRev.133.A224">Ising model and self-avoiding walks on hypercubical lattices and high density expansions</a>, Phys. Rev. 133 (1964) A224-A239.
%H D. MacDonald, D. L. Hunter, K. Kelly, and N. Jan, <a href="http://dx.doi.org/10.1088/0305-4470/25/6/006">Self-avoiding walks in two to five dimensions: exact enumerations and series study</a>, J Phys A: Math Gen 25 (1992) Vol. 6, 1429-1440 [Gives 18 terms]
%H A. M. Nemirovsky et al., <a href="http://dx.doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108.
%H Hugo Pfoertner, <a href="http://www.randomwalk.de/stw4d.html">Results for the 4D Self-Trapping Random Walk</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Self-AvoidingWalkConnectiveConstant.html">Self-Avoiding Walk Connective Constant</a>
%F a(n) = 8*A366925(n) for n >= 1. - _Hugo Pfoertner_, Nov 03 2023
%o A "brute force" FORTRAN program to count the 4D walks is available at the Pfoertner link.
%Y Cf. A001411, A001412, A242355, A323856, A323857, A366925.
%K nonn,walk,nice
%O 0,2
%A _N. J. A. Sloane_
%E a(12)-a(16) from _Hugo Pfoertner_, Dec 14 2002
%E a(17) from _Hugo Pfoertner_, Oct 19 2004
%E a(18) onwards from _R. J. Mathar_ using data from Clisby et al, Aug 31 2007
|
