OFFSET
0,2
COMMENTS
Fisher and Hiley give 396204 as their last term instead of 396172 (see A002932). Douglas McNeil confirms 396172 (see seqfan discussion).
Comment from N. J. A. Sloane, Nov 27 2010: Joseph Myers has discovered that several of the sequences listed by Fisher and Hiley (1961) contained errors. R. J. Mathar comments that this article has 62 citations in http://adsabs.harvard.edu/abs/1961JChPh..34.1253F and that clicking through these with the "Citations to the Article (62)" button is one way to check the numbers by searching for corrections.
From Petros Hadjicostas, Jan 01 2019: (Start)
Nemirovsky et al. (1992), for a d-dimensional hypercubic lattice, define C_{n,m} to be "the number of configurations of an n-bond self-avoiding chain with m neighbor contacts." For d=2 (square lattice) and m=0 (no neighbor contacts), we have (for the current sequence) a(n) = C(n, m=0). These values (from n=1 to n=11) are listed in Table I (p. 1088) in the paper.
According to Eq. (5), p. 1090, in the above paper, for a general d, the partition number C_{n,m} satisfies C_{n,m} = Sum_{l=1..n} 2^l*l!*Bin(d,l)*p_{n,m}^{(l)}, where the coefficients p_{n,m}^{(l)} (l=1,2,...) are independent of d. For d=2 (square lattice), this becomes C_{n,m} = Sum_{l=1..n} 2^l*l!*Bin(2,l)*p_{n,m}^{(l)}.
According to Eq. (7a) and (7b), p. 1093, in the paper, p_{n,0}^{(1)} = 1 = p_{n,0}^{(n)}, p_{n,m}^{(1)} = 0 for m >= 1, and p_{n,m}^{(l)} = 0 for m >= 1 and n-m+1 <= l <= n.
Now, assume d=2. Since p_{n,0}^{(1)} = 1 for n >= 1, we have C_{1,0} = 2^1*1!*Bin(2,1)*1 = 4, while C_{n,0} = 4 + 2^2*2!*Bin(2,2)*p_{n,0}^{(2)} = 4 + 8*p_{n,0}^{(2)} for n >= 2. The partition numbers p_{n,0}^{(2)} appear in Table II, p. 1093, in the paper. We have p_{n,0}^{(2)} = A038746(n) (with p_{1,0}^{(2)} = 0 to make the formula C_{n,0} = 4 + 8*p_{n,0}^{(2)} valid even for n=1). (End)
From Carlo Corti, Aug 11 2026: (Start)
Previous name was: Number of n-step walks on square lattice (no points repeated, no adjacent points unless consecutive in path).
The ancillary file sq_isaw_counts.ser accompanying Beaton, Guttmann, and Jensen (2020) contains the values c_{n,m}, the number of n-step self-avoiding walks on the square lattice with m nearest-neighbor contacts, for n <= 59. In this notation, a(n) = c_{n,0}. (End)
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Carlo Corti, Table of n, a(n) for n = 0..59 (original table for n = 0..35 from Scott R. Shannon)
N. R. Beaton, A. J. Guttmann, and I. Jensen, Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents, arXiv:1911.05852 [cond-mat.stat-mech], 2020.
D. Bennett-Wood, I. G. Enting, D. S. Gaunt, A. J. Guttmann, J. L. Leask, A. L. Owczarek, and S. G. Whittington, Exact enumeration study of free energies of interacting polygons and walks in two dimensions, J. Phys. A: Math. Gen. 31 (1998), 4725-4741. [See Table B1 (pp. 4738-4739), where the numbers must be multiplied by 4. - Petros Hadjicostas, Jan 05 2019]
M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.
A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Sequence Fans Mailing list, discussion of this sequence, November 2010
FORMULA
a(n) = 4 + 8*A038746(n) for n>=1.
a(n) = 2 * A182644(n+1) for n>=1. - Andrei Zabolotskii, Apr 08 2026
EXAMPLE
From Carlo Corti, Aug 11 2026: (Start)
a(3) = 28. There are A001411(3) = 36 three-step self-avoiding walks starting at a fixed lattice vertex. Exactly 8 of them go around three sides of a unit square and have adjacent endpoints, so they are excluded.
A representative, where the dotted segment between S and X is the forbidden contact and is not a step of the walk, is
S---o
. |
X---o
Thus a(3) = 36 - 8 = 28. (End)
MATHEMATICA
a[n_] := If[n == 0, 1, 8 A038746[[n]] + 4];
a /@ Range[0, 32] (* Jean-François Alcover, Feb 24 2020 *)
CROSSREFS
Cf. A038746 (corresponding symmetry classes after excluding straight walks), A182644 (fixed snake polyominoes), A001411 (all square-lattice self-avoiding walks), A033155 (d=2, m=1), A033323 (d=2, m=2), A336492 (total number of contacts).
KEYWORD
nonn,walk,nice
AUTHOR
Joseph Myers, Nov 22 2010
EXTENSIONS
a(23)-a(32) from Bert Dobbelaere, Jan 02 2019
a(33)-a(35) from Scott R. Shannon, Aug 25 2020
New name and a(36) from Carlo Corti, Aug 11 2026
STATUS
approved