A323063 -id:A323063 - cp's OEIS frontend

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

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

A038726 The number of n-step self-avoiding walks in a 5-dimensional hypercubic lattice with no non-contiguous adjacencies.

Original entry on oeis.org

1, 10, 90, 730, 5930, 47690, 384090, 3075610, 24663210, 197117210, 1576845050, 12589411530, 100567197770, 802350892730, 6403639865530

Offset: 0

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=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

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 Table I and Eq. 5 on p. 1090 (the case d=5).

Cf. A034006, A038746, A038748, A173380, A174319, A323037, A323063.

a(n) = 10 + 80*A038746(n) + 480*A038748(n) + 1920*A323037(n) + 3840*A323063(n). (It can be proved using Eq. (5), p. 1090, in the paper by Nemirovsky et al. (1992).) - Petros Hadjicostas, Jan 03 2019

Name edited by Petros Hadjicostas, Jan 02 2019

Title clarified, a(0), and a(12)-a(14) from Sean A. Irvine, Jul 29 2020

A323037 Coefficients arising in the enumeration of configurations of linear chains.

Original entry on oeis.org

0, 0, 0, 1, 13, 114, 849, 5842, 38174, 242737, 1511046

Offset: 1

Petros Hadjicostas, 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 = 4. This sequence appears in Table II, p. 1094 in the paper. (We have p_{n,0}^{(2)} = A038746(n) and p_{n,0}^{(3)} = A038748(n).)

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. A034006, A038746, A038748, A323063.

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

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A038726 The number of n-step self-avoiding walks in a 5-dimensional hypercubic lattice with no non-contiguous adjacencies.

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A323037 Coefficients arising in the enumeration of configurations of linear chains.

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