

A033155


Configurations of linear chains for a square lattice.


9



0, 0, 8, 32, 88, 256, 736, 2032, 5376, 14224, 36976, 95504, 243536, 619168, 1559168, 3916960, 9769072, 24321552, 60199464, 148803824, 366051864
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OFFSET

1,3


COMMENTS

From Petros Hadjicostas, Jan 03 2019: (Start)
In the notation of Nemirovsky et al. (1992), a(n), the nth term of the current sequence is C_{n,m} with m=1 (and d=2). Here, for a ddimensional hypercubic lattice, C_{n,m} is "the number of configurations of an nbond selfavoiding chain with m neighbor contacts."
These numbers are given in Table I (p. 1088) in the paper by Nemirovsky et al. (1992). Using Eqs. (5) and (7b) in the paper, we can prove that C_{n,m=1} = 2^1*1!*Bin(2,1)*p_{n,m=1}^{(1)} + 2^2*2!*Bin(2,2)*p_{n,m=1}^{(2)} = 0 + 8*p_{n,m=1}^{(2)} = 8*A038747(n).
(End)
The terms a(12) to a(21) were copied from Table B1 (pp. 47384739) in BennettWood et al. (1998). In the table, the authors actually calculate a(n)/4 = C(n, m=1)/4 for 1 <= n <= 29. (They use the notation c_n(k), where k stands for m, which equals 1 here. They call c_n(k) "the number of SAWs of length n with k nearestneighbour contacts".)  Petros Hadjicostas, Jan 04 2019


LINKS

Table of n, a(n) for n=1..21.
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.
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; see Eq. 5 (p. 1090) and Eq. 7b (p. 1093).


FORMULA

a(n) = 8*A038747(n) for n >= 1. (It can be proved using Eqs. (5) and (7b) in the paper by Nemirovsky et al. (1992).)  Petros Hadjicostas, Jan 03 2019


CROSSREFS

Cf. A038747.
Sequence in context: A018839 A008412 A014819 * A132117 A159941 A053348
Adjacent sequences: A033152 A033153 A033154 * A033156 A033157 A033158


KEYWORD

nonn,more


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Name edited by Petros Hadjicostas, Jan 03 2019


STATUS

approved



