A038746 - OEIS

%I #33 Jan 02 2019 11:39:31

%S 0,1,3,8,20,49,117,280,665,1583,3742,8876,20933,49521,116578,275204,

%T 646908,1524457,3579100,8421786,19752217,46419251,108774693,255351249,

%U 597911623,1402287934,3281303692,7689321700,17982126657,42108189097,98421806690,230322480772

%N Coefficients arising in the enumeration of configurations of linear chains.

%C This counts non-self-intersecting paths of length n on the square lattice, start and end points distinguished, straight line paths not counted, rotations and reflections of a path not counted as distinct from that path.

%C From _Petros Hadjicostas_, Jan 01 2019: (Start)

%C Nemirovsky et al. (1992), for a d-dimensional hypercubic lattice, define C_{n,m} to be "the number of configurations of an n-bond self-avoiding chain with m neighbor contacts." For d=2 (square lattice) and m=0 (no neighbor contacts), we have C(n, m=0) = A173380(n). These values (from n=1 to n=11) are listed in Table I (p. 1088) in the paper.

%C According to Eq. (5), p. 1090, in the above paper, for a general d, the partition number C_{n,m} satisfies C_{n,m} = Sum_{l=1..n} 2^l*l!*Bin(d,l)*p_{n,m}^{(l)}, where the coefficients p_{n,m}^{(l)} (l=1,2,...) are independent of d. For d=2 (square lattice), this becomes C_{n,m} = Sum_{l=1..n} 2^l*l!*Bin(2,l)*p_{n,m}^{(l)}.

%C According to Eq. (7a) and (7b), p. 1093, in the paper, p_{n,0}^{(1)} = 1 = p_{n,0}^{(n)}, p_{n,m}^{(1)} = 0 for m >= 1, and p_{n,m}^{(l)} = 0 for m >= 1 and n-m+1 <= l <= n.

%C Now, assume d=2. Since p_{n,0}^{(1)} = 1 for n >= 1, we have C_{1,0} = 2^1*1!*Bin(2,1)*1 = 4, while C_{n,0} = 4 + 2^2*2!*Bin(2,2)*p_{n,0}^{(2)} = 4 + 8*p_{n,0}^{(2)} for n >= 2. The partition numbers p_{n,0}^{(2)} appear in Table II, p. 1093, in the paper. For the current sequence, we have a(n) = p_{n,0}^{(2)} (with a(1) = p_{1,0}^{(2)} = 0 to make the formula A173380(n) = C_{n,0} = 4 + 8*p_{n,0}^{(2)} = 4 + 8*a(n) valid even for n=1).

%C Apparently, some of the numbers C_{n,m} (for d=2 and d=3) are calculated in Fisher and Hiley (1961); see Table II, p. 1261 (square and cubic). For d=2, they calculate C(n,0) for 1 <= n < 14, while for d=3, they calculate C(n,0) for 1 <= n <= 10. It seems, however, that there are some possible typos there. The typos (for both d=2 and d=3) become apparent if one compares their results with the numbers in Table I (p. 1088) in Nemirovsky et al. (1992). See the comments for the sequence A173380 for more details.

%C (End)

%C No adjacent points allowed unless consecutive in path - _Bert Dobbelaere_, Jan 02 2019

%H M. E. Fisher and B. J. Hiley, <a href="http://dx.doi.org/10.1063/1.1731729">Configuration and free energy of a polymer molecule with solvent interaction</a>, J. Chem. Phys., 34 (1961), 1253-1267.

%H A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, <a href="http://dx.doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108; see Eq. 5 (p. 1090).

%Y A173380(n) = 8*a(n) + 4.

%Y Cf. A002932, A002934, A033155, A033323, A034006, A042949, A046788, A047057, A174319.

%K nonn,walk,more

%O 1,3

%A _N. J. A. Sloane_, May 02 2000

%E Initial 0 added to match offset in reference, further explanation and terms a(12) = 8876 to a(22) = 46419251 by _Joseph Myers_, Nov 22 2010

%E a(23)-a(32) from _Bert Dobbelaere_, Jan 02 2019