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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 (list; graph; refs; listen; history; text; internal format)



From Petros Hadjicostas, Jan 03 2019: (Start)

In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence is C_{n,m} with m=1 (and d=2). Here, for a d-dimensional hypercubic lattice, C_{n,m} is "the number of configurations of an n-bond self-avoiding 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).


The terms a(12) to a(21) were copied from Table B1 (pp. 4738-4739) in Bennett-Wood 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 nearest-neighbour contacts".) - Petros Hadjicostas, Jan 04 2019


Table of n, a(n) for n=1..27.

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.

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.

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), 1083-1108; see Eq. 5 (p. 1090) and Eq. 7b (p. 1093).


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


Cf. A038747.

Sequence in context: A018839 A008412 A014819 * A132117 A159941 A053348

Adjacent sequences:  A033152 A033153 A033154 * A033156 A033157 A033158




N. J. A. Sloane.


Name edited by Petros Hadjicostas, Jan 03 2019

a(22)-a(27) from Sean A. Irvine, Jul 03 2020



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