A323857 -id:A323857 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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

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

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

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

A242355 Sum of squared end-to-end distances of all n-step self-avoiding walks on the 4-d cubic lattice.

Original entry on oeis.org

8, 128, 1416, 13568, 119960, 1009440, 8205656, 65068352, 506193144, 3879735776, 29378067080, 220265711040, 1637726387096, 12091336503584, 88727095777896, 647661676223168, 4705654523841704, 34049855885188128, 245482626441965048, 1764039730476165824

Offset: 1

Hugo Pfoertner, Aug 16 2014

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..24 [4th column in Table A6 from Clisby article 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.
N. Clisby, R. Liang and G. Slade Self-avoiding walk enumeration via the lace expansion. Tables in machine-readable format.
Hugo Pfoertner, Results for the 4D Self-Trapping Random Walk.

Cf. A010575 corresponding number of walks, A118313 end-to-end distances for cubic lattice, A078797 end-to-end distances for quadratic lattice, A323856, A323857.

A323856 Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.

Original entry on oeis.org

1, 16, 177, 1696, 14995, 126180, 1025707, 8133544, 63274143, 484966972, 3672258385, 27533213880, 204715798387, 1511417062948, 11090886972237, 80957709527896, 588206815480213, 4256231985648516, 30685328305245631, 220504966309520728, 1579874958814261407

Offset: 1

Hugo Pfoertner, Feb 03 2019

			a(1) = 1 is the square displacement of the fixed initial step.
a(2) = 16, because one of the A010575(2)/8 = 7 end points is (2,0,0,0) with square distance 4 and the other 6 end points (1,-1,0,0), (1,1,0,0), (1,0,-1,0), (1,0,1,0), (1,0,0,-1), (1,0,0,1) all have square distance 2. 16 = 1*4 + 6*2.
a(3) = 177, because there are 6 end points with square distance 1, e.g., (0,1,0,0), 24 end points with square distance 3, e.g., (1,1,1,0), 18 end points with square distance 5, e.g., (2,1,0,0), and 1 end point with square distance 9, (3,0,0,0). 177 = 6*1 + 24*3 + 18*5 + 1*9.

For references and links see A010575 and A242355.

Cf. A010575, A078605, A118313, A242355, A323857.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Fortran

Formula

Extensions

A242355 Sum of squared end-to-end distances of all n-step self-avoiding walks on the 4-d cubic lattice.

Views

Author

Keywords

Links

Crossrefs

A323856 Sum of square displacements over all self-avoiding n-step walks on 4-d cubic lattice with first step specified, A242355(n)/8.

Views

Author

Keywords

Examples

References

Crossrefs