Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #50 Jan 08 2019 03:01:56
%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="http://bit-player.org/wp-content/extras/bph-publications/AmSci-1998-07-Hayes-self-avoidance.pdf">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="http://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