A266549 - OEIS

A266549

Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

%I #39 May 25 2019 06:10:32

%S 0,1,1,3,6,25,86,414,1975,10479,56572,316577,1800363,10419605,

%T 61061169,361978851

%N Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

%C Differs from A057730 beginning at n = 8, since that sequence includes polyominoes with holes.

%H Joerg Arndt, <a href="/A266549/a266549.pdf">All a(6)=25 walks of length 12</a>, 2018

%H Brendan Owen, <a href="http://www.recmath.com/PolyPages/PolyPages/index.htm?Isopolyos.html">Isoperimetrical Polyominoes</a>, part of Andrew I. Clarke's Poly Pages.

%H Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A002931&name2=A266549&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true">Illustration of ratio A002931(n)/a(n) using Plot2</a>, showing apparent limit of 8.

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a216194.htm">Illustration of polygons of perimeter <= 16</a>.

%Y Apparently lim A002931(n)/a(n) = 8 for increasing n, accounting for (in most cases) 4 rotations times two flips. - _Joerg Arndt_, _Hugo Pfoertner_, Jul 09 2018

%Y Cf. A010566, A037245 (open self-avoiding walks), A316194.

%K nonn,hard,more,nice

%O 1,4

%A _Luca Petrone_, Dec 31 2015

%E a(11)-a(16) from _Joerg Arndt_, Jan 25 2018