cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A033155 Configurations of linear chains for a square lattice.

Original entry on oeis.org

0, 0, 8, 32, 88, 256, 736, 2032, 5376, 14224, 36976, 95504, 243536, 619168, 1559168, 3916960, 9769072, 24321552, 60199464, 148803824, 366051864, 899559584, 2201636848, 5384254000, 13121348672, 31957730688, 77595810512
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

From Petros Hadjicostas, Jan 03 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=2). 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."
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(2,1)*p_{n,m=1}^{(1)} + 2^2*2!*Bin(2,2)*p_{n,m=1}^{(2)} = 0 + 8*p_{n,m=1}^{(2)} = 8*A038747(n).
(End)
The terms a(12) to a(21) were copied from Table B1 (pp. 4738-4739) in Bennett-Wood et al. (1998). In the table, the authors actually calculate a(n)/4 = C(n, m=1)/4 for 1 <= n <= 29. (They use the notation c_n(k), where k stands for m, which equals 1 here. They call c_n(k) "the number of SAWs of length n with k nearest-neighbour contacts".) - Petros Hadjicostas, Jan 04 2019

Links

Crossrefs

Cf. A038747.

Formula

a(n) = 8*A038747(n) for n >= 1. (It can be proved using Eqs. (5) and (7b) in the paper by Nemirovsky et al. (1992).) - Petros Hadjicostas, Jan 03 2019

Extensions

Name edited by Petros Hadjicostas, Jan 03 2019
a(22)-a(27) from Sean A. Irvine, Jul 03 2020

A047057 Number of configurations of linear chains in a cubic lattice.

Original entry on oeis.org

0, 0, 24, 192, 1032, 5376, 26688, 128880, 605664, 2802576, 12755136, 57525552, 256574352, 1137418464, 5001796944, 21899428128, 95296531680, 413331190896
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

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)

Links

Crossrefs

Cf. A033155, A038747, A038748, A038749.

Formula

a(n) = 24*A038747(n) + 48*A038749(n) for n >= 1. - Petros Hadjicostas, Jan 04 2019

Extensions

Name edited by Petros Hadjicostas, Jan 04 2019
a(12)-a(18) from Sean A. Irvine, Jan 31 2021

A038749 Coefficients arising in the enumeration of configurations of linear chains.

Original entry on oeis.org

0, 0, 0, 2, 16, 96, 510, 2558, 12282, 57498, 263421, 1192480, 5330078, 23657520, 104106655, 455993276, 1984733843, 8609546380, 37164674383
Offset: 1

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Author

N. J. A. Sloane, May 02 2000

Keywords

Comments

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=3. These numbers are given in Table II (p. 1093) in the paper. This sequence can be used for the calculation of sequence A047057 via Eq. (5) in the paper by Nemirovsky et al. (1992). (Note that, by equations (7b) in the paper, p_{n,m=1}^{(1)} = 0 for all n >= 1. Also, p_{n,m=1}^{(2)} = A038747(n) for n >= 1.) - Petros Hadjicostas, Jan 04 2019

Links

Crossrefs

Cf. A033155, A038747, A047057.

Extensions

The first three 0's in the sequence were added by Petros Hadjicostas, Jan 04 2019 to make it agree with Table II (p. 1093) and Eq. (5) (p. 1090) in the paper by Nemirovsky et al. (1992).
a(12)-a(19) from Sean A. Irvine, Feb 02 2021

A336492 Total number of neighbor contacts for n-step self-avoiding walks on a 2D square lattice.

Original entry on oeis.org

0, 0, 8, 32, 152, 512, 1880, 5920, 19464, 59168, 183776, 545392, 1638400, 4778000, 14043224, 40422544, 116977176, 333346928, 953538440, 2695689520, 7642091352, 21464794032, 60417010152, 168787016352, 472315518008, 1313548558528, 3657850909680, 10133559518800
Offset: 1

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Author

Scott R. Shannon, Jul 23 2020

Keywords

Comments

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.

Examples

			a(1) = a(2) = 0 as a 1 and 2 step walk cannot approach a previous step.
a(3) = 8. The single walk where one interaction occurs, which can be taken in eight ways on a 2D square lattice, is:
.
   +---+
       |
   X---+
.
Therefore, the total number of interactions is 1*1*8 = 8.
a(4) = 32. The four walks where one interaction occurs, each of which can be taken in eight ways on a 2D square lattice, are:
.
  +---+---+   +           +---+       +---+
          |   |               |       |   |
      X---+   +---+   X---+---+   X---+   +
                  |
              X---+
.
Therefore, the total number of interactions is 4*1*8 = 32.
a(5) = 152. Considering only walks which start with one or more steps to the right followed by an upward step there are thirty-five different walks. Eleven of these have one neighbor contact (hence A033155(5) = 11*8 = 88) while four have two contacts. These are:
.
  +---+---+   +---+---+   +---+   +---+
  |       |           |   |       |   |
  +   X---+   X---+---+   +---+   +   +
                              |       |
                          X---+   X---+
.
Therefore, the total number of contacts is (11*1 + 4*2)*8 = 152.

Links

Crossrefs

Cf. A033155 (total number of n-step walks containing one neighbor contact), A038747, A047057, A173380, A174319.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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