A033323 Configurations of linear chains in a square lattice.

0, 0, 0, 0, 32, 128, 344, 1072, 3400, 9832, 27600, 77000, 211736, 572560, 1534512, 4072664, 10725424, 28035128, 72831272, 188139616, 483452824, 1236865976, 3150044696, 7994665480, 20209319824, 50942982080

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

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=2 (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 appear in Table I (p. 1088) in the paper by Nemirovsky et al. (1992).

(End)

The terms a(12) to a(19) 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=2)/4 for 1 <= n <= 29. (They use the notation c_n(k), where k stands for m, which equals 2 here. They call c_n(k) "the number of SAWs of length n with k nearest-neighbour contacts".) - Petros Hadjicostas, Jan 04 2019

Links

• 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.
• 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).

Extensions

Name edited by and more terms from Petros Hadjicostas, Jan 03 2019

a(20)-a(26) from Sean A. Irvine, Jul 03 2020