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A010575 Number of n-step self-avoiding walks on 4-d cubic lattice. 4

%I

%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 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 link below]

%H N. Clisby, R. Liang and G. Slade <a href="http://dx.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 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>

%o A "brute force" FORTRAN program to count the 4D walks is available at the Pfoertner link.

%K nonn,walk,nice

%O 0,2

%A _N. J. A. Sloane_.

%E Extended to n=16. 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

%E More terms from _Hugo Pfoertner_, Dec 14 2002; Oct 19 2004

%E Further terms from _R. J. Mathar_, Aug 31 2007

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Last modified March 24 02:10 EDT 2017. Contains 283984 sequences.