A038747 Coefficients arising in the enumeration of configurations of linear chains.

0, 0, 1, 4, 11, 32, 92, 254, 672, 1778, 4622, 11938, 30442, 77396, 194896, 489620, 1221134, 3040194, 7524933, 18600478, 45756483, 112444948, 275204606, 673031750, 1640168584, 3994716336, 9699476314

Offset: 1

Author

N. J. A. Sloane, May 02 2000

Keywords

• nonn
• more

Comments

In the notation of Nemirovsky et al. (1992), a(n), the n-th term of this sequence is p_{n,m}^{(l)} with m=1 and l=2. These numbers are given in Table II (p. 1093) in the paper. This sequence can be used for the calculation of sequence A033155(n) via Eq. (5) in the paper by Nemirovsky et al. (1992). (Note that, by equations (7b) in the paper, p_{n,1}^{(1)} = 0 for all n >= 1.) - Petros Hadjicostas, Jan 03 2019

In Table B1 (pp. 4738-4739), Bennett-Wood et al. (1998) tabulated c_n(k)/4, for various values of n and k, where c_n(k) is "the number of SAWs of length n with k nearest-neighbour contacts". (Here, the letter k stands for the letter m in the previous paragraph.) Bennett-Wood et al. (1998) worked only with a square lattice (i.e., d=2) unlike Nemirovsky et al. (1992) who worked with a d-dimensional hypercubic lattice. Both papers deal with SAWs = self-avoiding walks (in a lattice). We have c_n(k=1) = A033155(n) = 8*p_{n,1}^{(2)}, i.e., a(n) = p_{n,1}^{(2)} = (c_n(k=1)/4)/2, and this is the reason the numbers in Table B1 in Bennett-Wood et al. (1998) must be divided by 2 in order to get extra terms for the current sequence (a(12) to a(24)). - Petros Hadjicostas, Jan 05 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).

Crossrefs

Cf. A033155, A038749, A047057.

Extensions

The first two 0's in the sequence were inserted by Petros Hadjicostas, Jan 03 2019 to make it agree with Table II (p. 1093) and Eq. (5) (p. 1090) in the paper by Nemirovsky et al. (1992)

Terms a(12) to a(24) were copied from Table II, p. 4738, in the paper by Bennett-Wood et al. (1998) (after division by 2) by Petros Hadjicostas, Jan 05 2019

a(25)-a(27) from Sean A. Irvine, Jul 03 2020