

A173380


Number of nstep walks on square lattice (no points repeated, no adjacent points unless consecutive in path).


16



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
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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 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 ddimensional hypercubic lattice, define C_{n,m} to be "the number of configurations of an nbond selfavoiding 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 nm+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. BennettWood, 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), 47254741. [See Table B1 (pp. 47384739), 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), 12531267.
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), 10831108.
Sequence Fans Mailing list, discussion of this sequence, November 2010


FORMULA

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


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] (* JeanFrançois Alcover, Feb 24 2020 *)


CROSSREFS

Cf. A002932, A002934, A033155, A033323, A034006, A038746, A042949, A046788, A047057, A174319.
Sequence in context: A317233 A309917 A034508 * A002932 A337441 A242079
Adjacent sequences: A173377 A173378 A173379 * A173381 A173382 A173383


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


STATUS

approved



