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A033155
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Configurations of linear chains for a square lattice.
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11
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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
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OFFSET
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1,3
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COMMENTS
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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).
(End)
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
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LINKS
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FORMULA
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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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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