login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A010575 Number of n-step self-avoiding walks on 4-d cubic lattice. 13
1, 8, 56, 392, 2696, 18584, 127160, 871256, 5946200, 40613816, 276750536, 1886784200, 12843449288, 87456597656, 594876193016, 4047352264616, 27514497698984, 187083712725224, 1271271096363128, 8639846411760440, 58689235680164600, 398715967140863864 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant
FORMULA
a(n) = 8*A366925(n) for n >= 1. - Hugo Pfoertner, Nov 03 2023
PROG
A "brute force" FORTRAN program to count the 4D walks is available at the Pfoertner link.
CROSSREFS
Sequence in context: A001666 A214942 A010556 * A162949 A063812 A234274
KEYWORD
nonn,walk,nice
AUTHOR
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
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 07:04 EDT 2024. Contains 370953 sequences. (Running on oeis4.)