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

Author

Bert Dobbelaere, Jan 06 2019

Keywords

• nonn
• walk

Comments

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).

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

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