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 A047057 Number of configurations of linear chains in a cubic lattice. 7
 0, 0, 24, 192, 1032, 5376, 26688, 128880, 605664, 2802576, 12755136 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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. FORMULA a(n) = 24*A038747(n) + 48*A038749(n) for n >= 1. - Petros Hadjicostas, Jan 04 2019 CROSSREFS Cf. A033155, A038747, A038748, A038749. Sequence in context: A143040 A305166 A128960 * A066406 A271432 A042114 Adjacent sequences:  A047054 A047055 A047056 * A047058 A047059 A047060 KEYWORD nonn,more AUTHOR EXTENSIONS Name edited by Petros Hadjicostas, Jan 04 2019 STATUS approved

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Last modified October 20 22:05 EDT 2019. Contains 328291 sequences. (Running on oeis4.)