A323188 -id:A323188 - cp's OEIS frontend

A037245 Number of unrooted self-avoiding walks of n steps on square lattice.

1, 2, 4, 9, 22, 56, 147, 388, 1047, 2806, 7600, 20437, 55313, 148752, 401629, 1078746, 2905751, 7793632, 20949045, 56112530, 150561752, 402802376, 1079193821, 2884195424, 7717665979, 20607171273, 55082560423, 146961482787, 392462843329, 1046373230168, 2792115083878

Offset: 1

Brian Hayes

Or, number of 2-sided polyedges with n cells. - Ed Pegg Jr, May 13 2009
A walk and its reflection (i.e., exchange start and end of walk, what Hayes calls a "retroreflection") are considered one and the same here. - Joerg Arndt, Jan 26 2018
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
Joerg Arndt, The a(6) = 56 walks of length 6, 2018 (pdf, 2 pages).
Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyedge

Asymptotically approaches (1/16) * A001411.
Cf. A266549 (closed self-avoiding walks).
Cf. A323188, A323189 (program).

a(n) = (A001411(n) + A323188(n) + A323189(n) + 4) / 16. - Bert Dobbelaere, Jan 07 2019

a(25)-a(27) from Luca Petrone, Dec 20 2015

More terms using formula by Bert Dobbelaere, Jan 07 2019

A316194 Number of symmetric self-avoiding polygons on square lattice with perimeter 2*n, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 1, 1, 3, 4, 16, 23, 87

Offset: 1

Hugo Pfoertner, Jun 27 2018

The sequence includes polygons of 2-fold, i.e., mirror or rotational, and higher (order >= 4) symmetry.

Hugo Pfoertner, Illustration of polygons of perimeter <= 16.

Cf. A002931, A266549, A316196, A323188, A323189.

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.

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A037245 Number of unrooted self-avoiding walks of n steps on square lattice.

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A316194 Number of symmetric self-avoiding polygons on square lattice with perimeter 2*n, not counting rotations and reflections as distinct.

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

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