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

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 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.

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

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

Adjacent sequences:  A010572 A010573 A010574 * A010576 A010577 A010578

KEYWORD

nonn,walk,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

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

More terms from Hugo Pfoertner, Dec 14 2002; Oct 19 2004

Further terms from R. J. Mathar, Aug 31 2007

STATUS

approved

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Last modified August 18 04:17 EDT 2017. Contains 290684 sequences.