A002934 -id:A002934 - cp's OEIS frontend

A173380 Number of n-step walks on square lattice (no points repeated, no adjacent points unless consecutive in path).

1, 4, 12, 28, 68, 164, 396, 940, 2244, 5324, 12668, 29940, 71012, 167468, 396172, 932628, 2201636, 5175268, 12195660, 28632804, 67374292, 158017740, 371354012, 870197548, 2042809996, 4783292988, 11218303476, 26250429540, 61514573604, 143857013260, 336865512780, 787374453524, 1842579846180

Offset: 0

Joseph Myers, Nov 22 2010

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).
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Scott R. Shannon, Table of n, a(n) for n = 0..35
D. Bennett-Wood, I. G. Enting, D. S. Gaunt, A. J. Guttmann, J. L. Leask, A. L. Owczarek, and S. G. Whittington, Exact enumeration study of free energies of interacting polygons and walks in two dimensions, J. Phys. A: Math. Gen. 31 (1998), 4725-4741. [See Table B1 (pp. 4738-4739), where the numbers must be multiplied by 4. - _Petros Hadjicostas_, Jan 05 2019]
M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.
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.
Sequence Fans Mailing list, discussion of this sequence, November 2010

Cf. A002932, A002934, A033155, A033323, A034006, A038746, A042949, A046788, A047057, A174319.

A038746 = Cases[Import["https://oeis.org/A038746/b038746.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 1, 8 A038746[[n]] + 4];
a /@ Range[0, 32] (* Jean-François Alcover, Feb 24 2020 *)

a(n) = 4 + 8*A038746(n) for n>=1.

a(23)-a(32) from Bert Dobbelaere, Jan 02 2019

a(33)-a(35) from Scott R. Shannon, Aug 25 2020

A174319 Number of n-step walks on cubic lattice (no points repeated, no adjacent points unless consecutive in path).

Original entry on oeis.org

1, 6, 30, 126, 534, 2214, 9246, 38142, 157974, 649086, 2675022, 10966470, 45054630, 184400910, 755930958, 3089851782, 12645783414, 51635728518, 211059485310, 861083848998, 3516072837894, 14334995983614, 58485689950254

Offset: 0

Joseph Myers, Nov 27 2010

Fisher and Hiley give 2674926 as their last term instead of 2675022 (see A002934). Douglas McNeil confirms the correction on the seqfan list.

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=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." (Let n >= 1. For d=2, we have C(n,0) = A173380(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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.
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.

Cf. A002934, A038746, A038748.

a(n) = 6 + 24*A038746(n) + 48*A038748(n) for n >= 1. (It follows from Eq. (5), p. 1090, in Nemirovsky et al. (1992).) - Petros Hadjicostas, Jan 01 2019

a(16)-a(22) from Bert Dobbelaere, Jan 03 2019

A038746 Coefficients arising in the enumeration of configurations of linear chains.

Original entry on oeis.org

0, 1, 3, 8, 20, 49, 117, 280, 665, 1583, 3742, 8876, 20933, 49521, 116578, 275204, 646908, 1524457, 3579100, 8421786, 19752217, 46419251, 108774693, 255351249, 597911623, 1402287934, 3281303692, 7689321700, 17982126657, 42108189097, 98421806690, 230322480772

Offset: 1

N. J. A. Sloane, May 02 2000

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.
(End)
No adjacent points allowed unless consecutive in path - Bert Dobbelaere, Jan 02 2019

M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.
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; see Eq. 5 (p. 1090).

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

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

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

a(23)-a(32) from Bert Dobbelaere, Jan 02 2019

A038748 Coefficients arising in the enumeration of configurations of linear chains.

Original entry on oeis.org

0, 0, 1, 7, 36, 168, 736, 3151, 13190, 54938, 226597, 934200, 3831219, 15723801, 64313623, 263316219, 1075420890, 4396310382, 17937457304, 73247306563, 298635873550, 1218428664338

Offset: 1

N. J. A. Sloane, May 02 2000

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

			From _Bert Dobbelaere_, Jan 03 2019: (Start)
Using strings to represent a path with characters X,Y,Z for steps in positive directions and x,y,z for steps in negative directions along the respective axes, the following enumerations correspond to the first nonzero terms:
a(3) = 1: { XYZ }
a(4) = 7: { XXYZ, XYXZ, XYYZ, XYZX, XYZx, XYZY, XYZZ }
a(5) = 36: {
      XXXYZ, XXYXZ, XXYYZ, XXYZX, XXYZx, XXYZY, XXYZZ, XYXXZ, XYXYZ,
      XYXZX, XYXZY, XYXZy, XYXZZ, XYYXZ, XYYxZ, XYYYZ, XYYZX, XYYZx,
      XYYZY, XYYZZ, XYZXX, XYZXY, XYZXy, XYZXZ, XYZxx, XYZxY, XYZxZ,
      XYZYX, XYZYx, XYZYY, XYZYZ, XYZZX, XYZZx, XYZZY, XYZZy, XYZZZ }
Symmetries are avoided by imposing the following restrictions: all patterns start with 'X'. First occurrence of 'Y' comes before the first occurrence of 'Z' (presence mandatory). First occurrence of steps in negative directions (presence optional) comes after the first occurrence of the corresponding steps in positive directions.
(End)

M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.
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; see Eq. 5 (p. 1090).

Cf. A002934, A174319, A323063.

Terms a(12) to a(15) were calculated by Petros Hadjicostas, Jan 01 2019 using Eq. (5) in Nemirovsky et al. (1992) and the terms of the sequences A038746 and A174319.

a(12)-a(15) confirmed by direct computation and a(16)-a(22) from Bert Dobbelaere, Jan 03 2019

A038729 Configurations of linear chains in a 6-dimensional hypercubic lattice.

Original entry on oeis.org

12, 132, 1332, 13452, 134892, 1353732, 13536612, 135457932, 1352852292, 13517235732, 134908128732, 1346796414252, 13435850843172

Offset: 1

N. J. A. Sloane, May 02 2000

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

M. E. Fisher and B. J. Hiley, Configuration and free energy of a polymer molecule with solvent interaction, J. Chem. Phys., 34 (1961), 1253-1267.
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; see Eq. 5 (p. 1090) and Table 1 (p. 1090).

Cf. A002932, A002934, A034006, A038726, A173380, A174313, A174319.

Terms a(10) and a(11) were copied from Table 1 (p. 1090) in the paper by Nemirovsky et al. (1992) by Petros Hadjicostas, Jan 03 2019
Name edited by Petros Hadjicostas, Jan 03 2019
a(12)-a(13) from Sean A. Irvine, Feb 01 2021

cp's OEIS Frontend

A173380 Number of n-step walks on square lattice (no points repeated, no adjacent points unless consecutive in path).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A174319 Number of n-step walks on cubic lattice (no points repeated, no adjacent points unless consecutive in path).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A038746 Coefficients arising in the enumeration of configurations of linear chains.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A038748 Coefficients arising in the enumeration of configurations of linear chains.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A038729 Configurations of linear chains in a 6-dimensional hypercubic lattice.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions