cp's OEIS Frontend

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)

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