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

1, 8, 56, 392, 2696, 18584, 127160, 871256, 5946200, 40613816, 276750536, 1886784200, 12843449288, 87456597656, 594876193016, 4047352264616, 27514497698984, 187083712725224, 1271271096363128, 8639846411760440, 58689235680164600, 398715967140863864

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• nice

Comments

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

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

Links

• 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

Crossrefs

Cf. A001411, A001412, A242355, A323856, A323857, A366925.

Programs

Fortran

c A "brute force" Fortran program to count the 4D walks is available at the Pfoertner link.

Formula

a(n) = 8*A366925(n) for n >= 1. - Hugo Pfoertner, Nov 03 2023

Extensions

a(12)-a(16) from Hugo Pfoertner, Dec 14 2002

a(17) from Hugo Pfoertner, Oct 19 2004

a(18) onwards from R. J. Mathar using data from Clisby et al, Aug 31 2007