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Fisher and Hiley give 396204 as their last term instead of 396172 (see A002932). Douglas McNeil confirms 396172 (see seqfan discussion).

Comment from N. J. A. Sloane, Nov 27 2010: Joseph Myers has discovered that several of the sequences listed by Fisher and Riley (1961) contained errors. R. J. Mathar comments that this article has 62 citations in http://adsabs.harvard.edu/abs/1961JChPh..34.1253F and that clicking through these with the "Citations to the Article (62)" button is one way to check the numbers by searching for corrections.

From Petros Hadjicostas, Jan 01 2019: (Start)

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 (for the current sequence) a(n) = C(n, m=0). These values (from n=1 to n=11) are listed in Table I (p. 1088) in the paper.

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)}.

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.

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. We have p_{n,0}^{(2)} = A038746(n) (with p_{1,0}^{(2)} = 0 to make the formula C_{n,0} = 4 + 8*p_{n,0}^{(2)} valid even for n=1).

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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=0 (and d=4). 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,0) = A173380(n), while for d=3, we have C(n,0) = A174319(n).) - Petros Hadjicostas, Jan 02 2019

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.

From Petros Hadjicostas, Jan 01 2019: (Start)

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.

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)}.

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.

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).

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.

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No adjacent points allowed unless consecutive in path - Bert Dobbelaere, Jan 02 2019

In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence, is equal to p_{n,m}^{(l)} with m = 0 and l = 3. - Petros Hadjicostas, Jan 02 2019

This counts non-self-intersecting paths of length n on the cubic lattice, start and end points distinguished, planar paths not counted, rotations and reflections of a path not counted as distinct from that path. No points repeated, no adjacent points allowed unless consecutive in path. - Bert Dobbelaere, Jan 03 2019

In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence is C_{n,m} with m=0 (and d=5). 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,0) = A173380(n); for d=3, we have C(n,0) = A174319(n); and for d=4, we have C(n,0) = A034006(n).) - Petros Hadjicostas, Jan 02 2019

In the notation of Nemirovsky et al. (1992), a(n), the n-th term of the current sequence is C_{n,m} with m=0 (and d=6). 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,0) = A173380(n); for d=3, we have C(n,0) = A174319(n); for d=4, we have C(n,0) = A034006(n); and for d=5, we have C(n,0) = A038726(n).) - Petros Hadjicostas, Jan 03 2019

This sequence gives the total number of neighbor contacts for all n-step self avoiding walks on a 2D square lattices. A neighbor contact is when the walk comes within 1 unit distance of a previously visited point, excluding the previous adjacent point.