A037245 -id:A037245 - cp's OEIS frontend

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

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.

A151538 Number of 1-sided strip polyedges with n cells.

Original entry on oeis.org

1, 2, 6, 14, 40, 102, 284, 752, 2069, 5547, 15134, 40712, 110456, 297066, 802808, 2156378, 5810329, 15584271, 41894990, 112217372, 301115391, 805584175, 2158366236, 5768337730, 15435275815, 41214200699, 110164972820, 293922598172, 784925297952, 2092745480990, 5584229143243

Offset: 1

Ed Pegg Jr, May 13 2009

With A001411 as main input and counting the symmetrical shapes separately, higher terms can be computed efficiently (see formula). - Bert Dobbelaere, Jan 07 2019

Bert Dobbelaere, Table of n, a(n) for n = 1..60
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyedge

Cf. A001411, A019988, A037245, A151537, A323189 (program).

a(n) = (A001411(n) + A323189(n)) / 8. - Bert Dobbelaere, Jan 07 2019

a(13)-a(19) from Joseph Myers, Oct 03 2011

More terms using formula by Bert Dobbelaere, Jan 07 2019

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

A002976 Number of certain self-avoiding walks with n steps on square lattice (see reference for precise definition).

Original entry on oeis.org

0, 1, 0, 2, 0, 5, 9, 21, 42, 76, 174, 396, 888, 2023, 4345, 9921, 22566

Offset: 4

N. J. A. Sloane

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

W. A. Beyer, Letter to N. J. A. Sloane, 1980
W. A. Beyer and M. B. Wells, Lower bound for the connective constant of a self-avoiding walk on a square lattice, J. Combin. Theory, A 13 (1972), 176-182.

Cf. A001411, A037245.

a(n) = A006142(n)+2*A006143(n)+A006144(n). - R. J. Mathar, Oct 22 2007

A151537 Number of 1-sided polyedges with n edges.

Original entry on oeis.org

1, 2, 7, 25, 99, 416, 1854, 8411, 38980, 182829, 867096, 4145168, 19955321, 96619260, 470157772

Offset: 1

Ed Pegg Jr, May 13 2009

Ishino Keiichiro, Polyedge, Puzzle will be played, 2009.
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyedge

Cf. A019988, A096267, A037245, A151538, A333249.

a(11)-a(14) from Joseph Myers, Oct 03 2011

a(15) from Ishino Keiichiro's website added by Andrey Zabolotskiy, Dec 10 2023

cp's OEIS Frontend

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

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A151538 Number of 1-sided strip polyedges with n cells.

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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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A002976 Number of certain self-avoiding walks with n steps on square lattice (see reference for precise definition).

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A151537 Number of 1-sided polyedges with n edges.

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