A047057 Number of configurations of linear chains in a cubic lattice.

0, 0, 24, 192, 1032, 5376, 26688, 128880, 605664, 2802576, 12755136, 57525552, 256574352, 1137418464, 5001796944, 21899428128, 95296531680, 413331190896

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

From Petros Hadjicostas, Jan 04 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=3). 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." (For d=2, we have C_{n,m=1} = A033155(n).)

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(3,1)*p_{n,m=1}^{(1)} + 2^2*2!*Bin(3,2)*p_{n,m=1}^{(2)} + 2^3*3!*Bin(3,3)*p_{n,m=1}^{(3)} = 0 + 24*p_{n,m=1}^{(2)} + 48*p_{n,m=1}^{(3)} = 24*A038747(n) + 48*A038749(n).

For an explanation of the meaning of p_{n,m}^{(l)} (l = 1,2,3,...), see the discussion that follows Eq. (5) in Nemirovsky et al. (1992), pp. 1090-1093. See also the comments for sequence A038748 by Bert Dobbelaere. (End)

Links

• 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.

Crossrefs

Cf. A033155, A038747, A038748, A038749.

Formula

a(n) = 24*A038747(n) + 48*A038749(n) for n >= 1. - Petros Hadjicostas, Jan 04 2019

Extensions

Name edited by Petros Hadjicostas, Jan 04 2019

a(12)-a(18) from Sean A. Irvine, Jan 31 2021