A037245 - OEIS

%I #53 Feb 26 2025 08:02:59

%S 1,2,4,9,22,56,147,388,1047,2806,7600,20437,55313,148752,401629,

%T 1078746,2905751,7793632,20949045,56112530,150561752,402802376,

%U 1079193821,2884195424,7717665979,20607171273,55082560423,146961482787,392462843329,1046373230168,2792115083878

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

%C Or, number of 2-sided polyedges with n cells. - _Ed Pegg Jr_, May 13 2009

%C 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

%C With A001411 as main input and counting the symmetrical shapes separately, higher terms can be computed efficiently (see formula). - _Bert Dobbelaere_, Jan 07 2019

%H Bert Dobbelaere, <a href="/A037245/b037245.txt">Table of n, a(n) for n = 1..60</a>

%H Joerg Arndt, <a href="/A037245/a037245.pdf">The a(6) = 56 walks of length 6</a>, 2018 (pdf, 2 pages).

%H Brian Hayes, <a href="https://www.americanscientist.org/article/how-to-avoid-yourself">How to avoid yourself</a>, American Scientist 86 (1998) 314-319.

%H Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/PolyformExplorer/">Illustrations of polyforms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyedge.html">Polyedge</a>

%F a(n) = (A001411(n) + A323188(n) + A323189(n) + 4) / 16. - _Bert Dobbelaere_, Jan 07 2019

%Y Asymptotically approaches (1/16) * A001411.

%Y Cf. A266549 (closed self-avoiding walks).

%Y Cf. A323188, A323189 (program).

%K nonn,walk,hard,nice

%O 1,2

%A _Brian Hayes_

%E a(25)-a(27) from _Luca Petrone_, Dec 20 2015

%E More terms using formula by _Bert Dobbelaere_, Jan 07 2019