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

%I A323189 #28 Feb 26 2025 08:02:39
%S A323189 4,4,12,12,36,36,100,100,284,276,780,764,2148,2084,5868,5692,15956,
%T A323189 15436,43300,41812,117100,112916,316076,304524,851612,819372,2290932,
%U A323189 2203132,6154284,5912572,16514988,15859820,44268460,42480972,118562580,113738396,317268516
%N A323189 Number of n-step point-symmetrical self-avoiding walks on the square lattice.
%C A323189 Total number of walks as counted in A001411 that have a point of symmetry.
%C A323189 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).
%H A323189 Bert Dobbelaere, <a href="/A323189/b323189.txt">Table of n, a(n) for n = 1..60</a>
%H A323189 Bert Dobbelaere, <a href="/A323189/a323189.pdf">Illustration of initial terms</a>
%H A323189 Bert Dobbelaere, <a href="/A323189/a323189.cpp.txt">C++ program</a>
%H A323189 Brian Hayes, <a href="https://www.americanscientist.org/article/how-to-avoid-yourself">How to avoid yourself</a>, American Scientist 86 (1998) 314-319.
%F A323189 A037245(n) = (A001411(n) + A323188(n) + a(n) + 4) / 16.
%F A323189 A151538(n) = (A001411(n) + a(n)) / 8.
%t A323189 A001411 = Import["https://oeis.org/A001411/b001411.txt", "Table"][[All, 2]];
%t A323189 A151538 = Import["https://oeis.org/A151538/b151538.txt", "Table"][[All, 2]];
%t A323189 a[n_] := 8 A151538[[n]] - A001411[[n+2]];
%t A323189 Array[a, 60] (* _Jean-François Alcover_, Sep 17 2019 *)
%Y A323189 Cf. A001411, A037245, A151538, A316194, A323188.
%K A323189 nonn,walk
%O A323189 1,1
%A A323189 _Bert Dobbelaere_, Jan 06 2019