A173380 Number of n-step walks on square lattice (no points repeated, no adjacent points unless consecutive in path).

1, 4, 12, 28, 68, 164, 396, 940, 2244, 5324, 12668, 29940, 71012, 167468, 396172, 932628, 2201636, 5175268, 12195660, 28632804, 67374292, 158017740, 371354012, 870197548, 2042809996, 4783292988, 11218303476, 26250429540, 61514573604, 143857013260, 336865512780, 787374453524, 1842579846180

Offset: 0

Author

Joseph Myers, Nov 22 2010

Keywords

• nonn
• walk
• nice

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 Riley (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)

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

• Scott R. Shannon, Table of n, a(n) for n = 0..35
• 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

Crossrefs

Cf. A002932, A002934, A033155, A033323, A034006, A038746, A042949, A046788, A047057, A174319.

Programs

Mathematica

A038746 = Cases[Import["https://oeis.org/A038746/b038746.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 1, 8 A038746[[n]] + 4];
a /@ Range[0, 32] (* Jean-François Alcover, Feb 24 2020 *)

Formula

a(n) = 4 + 8*A038746(n) for n>=1.

Extensions

a(23)-a(32) from Bert Dobbelaere, Jan 02 2019

a(33)-a(35) from Scott R. Shannon, Aug 25 2020