A316194 -id:A316194 - cp's OEIS frontend

A266549 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

0, 1, 1, 3, 6, 25, 86, 414, 1975, 10479, 56572, 316577, 1800363, 10419605, 61061169, 361978851

Offset: 1

Luca Petrone, Dec 31 2015

Differs from A057730 beginning at n = 8, since that sequence includes polyominoes with holes.

Joerg Arndt, All a(6)=25 walks of length 12, 2018
Brendan Owen, Isoperimetrical Polyominoes, part of Andrew I. Clarke's Poly Pages.
Hugo Pfoertner, Illustration of ratio A002931(n)/a(n) using Plot2, showing apparent limit of 8.
Hugo Pfoertner, Illustration of polygons of perimeter <= 16.

Apparently lim A002931(n)/a(n) = 8 for increasing n, accounting for (in most cases) 4 rotations times two flips. - Joerg Arndt, Hugo Pfoertner, Jul 09 2018

Cf. A010566, A037245 (open self-avoiding walks), A316194.

a(11)-a(16) from Joerg Arndt, Jan 25 2018

A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 27, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 30 2021

Closed walks are allowed.

			See link for illustrations of terms corresponding to diameters D < 8.5.

Hugo Pfoertner, Examples of paths of maximum length.

The squared radii of the enclosing circles are a subset of A192493/A192494.
Cf. A037245, A122224, A127399, A127400, A127401, A266549, A316194, A346993, A346994, A346995.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A323189 Number of n-step point-symmetrical self-avoiding walks on the square lattice.

Original entry on oeis.org

4, 4, 12, 12, 36, 36, 100, 100, 284, 276, 780, 764, 2148, 2084, 5868, 5692, 15956, 15436, 43300, 41812, 117100, 112916, 316076, 304524, 851612, 819372, 2290932, 2203132, 6154284, 5912572, 16514988, 15859820, 44268460, 42480972, 118562580, 113738396, 317268516

Offset: 1

Bert Dobbelaere, Jan 06 2019

Total number of walks as counted in A001411 that have a point of symmetry.

Note that for k > 4, we observe a(2k) < a(2k-1). This can be understood by considering interference between the parts at both sides of the point of symmetry (see illustration).

Bert Dobbelaere, Table of n, a(n) for n = 1..60
Bert Dobbelaere, Illustration of initial terms
Bert Dobbelaere, C++ program
Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.

Cf. A001411, A037245, A151538, A316194, A323188.

A001411 = Import["https://oeis.org/A001411/b001411.txt", "Table"][[All, 2]];
A151538 = Import["https://oeis.org/A151538/b151538.txt", "Table"][[All, 2]];
a[n_] := 8 A151538[[n]] - A001411[[n+2]];
Array[a, 60] (* Jean-François Alcover, Sep 17 2019 *)

A037245(n) = (A001411(n) + A323188(n) + a(n) + 4) / 16.

A151538(n) = (A001411(n) + a(n)) / 8.

A323188 Number of n-step mirror-symmetrical self-avoiding walks on the square lattice.

Original entry on oeis.org

4, 12, 12, 28, 28, 76, 76, 188, 196, 516, 524, 1292, 1356, 3500, 3596, 8908, 9380, 23940, 24796, 61500, 64900, 164612, 171244, 424940, 449140, 1134772, 1184204, 2939212, 3109644, 7834764, 8196100, 20345316, 21539420, 54156316, 56762036, 140908948, 149255908

Offset: 1

Bert Dobbelaere, Jan 06 2019

Total number of walks as counted in A001411 that have an axis of symmetry, either parallel to an axis or at a 45-degree angle (the latter only possible for even n).

Bert Dobbelaere, Table of n, a(n) for n = 1..60
Bert Dobbelaere, Illustration of initial terms
Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.

Cf. A001411, A037245, A151538, A316194, A323189 (program).

A037245 = Import["https://oeis.org/A037245/b037245.txt", "Table"][[All, 2]];
A151538 = Import["https://oeis.org/A151538/b151538.txt", "Table"][[All, 2]];
a[n_] := 16 A037245[[n]] - 8 A151538[[n]] - 4;
Array[a, 60] (* Jean-François Alcover, Sep 17 2019 *)

A037245(n) = (A001411(n) + a(n) + A323189(n) + 4) / 16.

A334756 Irregular table read by rows: T(n,k) is the number of 2n-step closed self-avoiding paths on a 2D square lattice with area k, where k >= n-1.

Original entry on oeis.org

0, 8, 24, 96, 16, 360, 160, 40, 1320, 960, 528, 144, 24, 4872, 4704, 3752, 2016, 840, 224, 56, 18112, 21632, 20992, 15424, 9920, 4832, 2176, 704, 192, 32, 67248, 96192, 107712, 93312, 75096, 50112, 31104, 16416, 7848, 3168, 1080, 288, 72

Offset: 1

Scott R. Shannon, May 10 2020

See A010566 for the number of closed self-avoiding 2D square lattice paths. Like that sequence here all possible paths are counted when determining the polygon areas, including those that are equivalent via rotation and reflection.

			For n = 2, total steps = 4, there are 8 different paths with an area of 1. These are the 8 possible ways to walk the polygon:
+---+
|   |
+---+
.
For n = 3, total steps = 6, there are 24 different paths with an area of 2. These are the 24 possible ways to walk the polygon:
+---+---+
|       |
+---+---+
.
For n = 4, total steps = 8, there are 96 different paths with an area of 3 and 16 different paths with an area of 4. These are the possible ways to walk the polygons:
+---+                      +---+---+
|   |                      |       |
+   +---+                  +       +
|       |                  |       |
+---+---+  for area = 3    +---+---+ for area = 4
.
For n = 5, total steps = 10, there are 360 different paths with an area of 4, 160 paths with an area of 5 and 40 different paths with an area of 6. These are the possible ways to walk the polygons:
+---+---+---+---+    +---+               +---+           +---+---+
|               |    |   |               |   |           |       |
+---+---+---+---+    +   +---+---+   +---+   +---+   +---+   +---+
                     |           |   |           |   |       |
                     +---+---+---+   +---+---+---+   +---+---+      for area = 4
.
+---+---+                      +---+---+---+
|       |                      |           |
+       +---+                  +           +
|           |                  |           |
+---+---+---+  for area = 5    +---+---+---+  for area = 6
.
Table begins:
0;
8;
24;
96,16;
360,160,40;
1320,960,528,144,24;
4872,4704,3752,2016,840,224,56;
18112,21632,20992,15424,9920,4832,2176,704,192,32;
67248,96192,107712,93312,75096,50112,31104,16416,7848,3168,1080,288,72;
249480,415040,526400,514480,468680,373280,281280,189920,120400,69120,36560,17040,7480,2720,880,240,40;
Row sums = A010566.

A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.
Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.
B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.
Iwan Jensen, Series Expansions for Self-Avoiding Walks
G. S. Rushbrooke and J. Eve, On Noncrossing Lattice Polygons, Journal of Chemical Physics, 31 (1959), 1333-1334.
Scott R. Shannon, Data for n=1..12.

Cf. A008855, A010566, A002931, A316194, A001411, A037245, A352838.

T(n, k) = 4 * n * A008855(k, n). - Andrey Zabolotskiy, Sep 27 2024

A316193 Number of symmetric self-avoiding polygons on honeycomb net with perimeter 2*n, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 10, 5

Offset: 1

Hugo Pfoertner, Jul 03 2018

The sequence includes polygons of 2-fold, i.e., mirror or rotational, and higher (order >= 3) symmetry.

Hugo Pfoertner, Illustration of polygons of perimeter <= 20.

Cf. A258206, A316194, A316196.

cp's OEIS Frontend

A266549 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

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A323189 Number of n-step point-symmetrical self-avoiding walks on the square lattice.

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A323188 Number of n-step mirror-symmetrical self-avoiding walks on the square lattice.

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A334756 Irregular table read by rows: T(n,k) is the number of 2n-step closed self-avoiding paths on a 2D square lattice with area k, where k >= n-1.

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A316193 Number of symmetric self-avoiding polygons on honeycomb net with perimeter 2*n, not counting rotations and reflections as distinct.

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