

A033155


Configurations of linear chains for a square lattice.


11



0, 0, 8, 32, 88, 256, 736, 2032, 5376, 14224, 36976, 95504, 243536, 619168, 1559168, 3916960, 9769072, 24321552, 60199464, 148803824, 366051864, 899559584, 2201636848, 5384254000, 13121348672, 31957730688, 77595810512
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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..27.
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.
Sean A. Irvine, Java program (github)
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
a(22)a(27) from Sean A. Irvine, Jul 03 2020


STATUS

approved



