A151538 -id:A151538 - 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.

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.

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

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A151537 Number of 1-sided polyedges with n edges.

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