cp's OEIS Frontend

A037245 Number of unrooted self-avoiding walks of n steps on square lattice.

%I A037245 #53 Feb 26 2025 08:02:59
%S A037245 1,2,4,9,22,56,147,388,1047,2806,7600,20437,55313,148752,401629,
%T A037245 1078746,2905751,7793632,20949045,56112530,150561752,402802376,
%U A037245 1079193821,2884195424,7717665979,20607171273,55082560423,146961482787,392462843329,1046373230168,2792115083878
%N A037245 Number of unrooted self-avoiding walks of n steps on square lattice.
%C A037245 Or, number of 2-sided polyedges with n cells. - _Ed Pegg Jr_, May 13 2009
%C A037245 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 A037245 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 A037245 Bert Dobbelaere, <a href="/A037245/b037245.txt">Table of n, a(n) for n = 1..60</a>
%H A037245 Joerg Arndt, <a href="/A037245/a037245.pdf">The a(6) = 56 walks of length 6</a>, 2018 (pdf, 2 pages).
%H A037245 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 A037245 Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/PolyformExplorer/">Illustrations of polyforms</a>
%H A037245 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyedge.html">Polyedge</a>
%F A037245 a(n) = (A001411(n) + A323188(n) + A323189(n) + 4) / 16. - _Bert Dobbelaere_, Jan 07 2019
%Y A037245 Asymptotically approaches (1/16) * A001411.
%Y A037245 Cf. A266549 (closed self-avoiding walks).
%Y A037245 Cf. A323188, A323189 (program).
%K A037245 nonn,walk,hard,nice
%O A037245 1,2
%A A037245 _Brian Hayes_
%E A037245 a(25)-a(27) from _Luca Petrone_, Dec 20 2015
%E A037245 More terms using formula by _Bert Dobbelaere_, Jan 07 2019