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A249565 Number of self-avoiding walks on the truncated square tiling with n steps.

%I A249565 #33 Oct 18 2024 11:43:09
%S A249565 1,3,6,12,22,42,80,152,284,536,988,1848,3412,6352,11724,21718,39952,
%T A249565 73808,135668,250188,459172,844888,1548608,2845186,5211548,9563768,
%U A249565 17501272,32079524,58660712,107425356,196320596,359232144,656099656,1199676412,2189995764
%N A249565 Number of self-avoiding walks on the truncated square tiling with n steps.
%C A249565 A self-avoiding walk is a sequence of adjacent points in a lattice that are all distinct.  The truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex.  The edge lattice is also referred to as (4,8^2) lattice.  It is also the Cayley graph of the Coxeter group generated by three generators {s_0, s_1, s_2} with the relations s_i^2 = 1, s_0 s_2 = s_2 s_0, (s_i s_{i+1})^4 = 1 for i=0,1.
%C A249565 It is conjectured that a(n) is approximately mu^n*n^{11/32} for large n where mu is the connective constant and mu is approximately 1.80883001(6).
%H A249565 Andrey Zabolotskiy, <a href="/A249565/b249565.txt">Table of n, a(n) for n = 0..47</a> (from Alm, 2005)
%H A249565 Sven Erick Alm, <a href="https://doi.org/10.1088/0305-4470/38/10/001">Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices</a>, J. Phys. A.: Math. Gen., 38 (2005), 2055-2080. Also <a href="https://citeseerx.ist.psu.edu/document?doi=17863725272f56f85b6ace259e9b8724f7db96b3">technical report</a> of the same name, 2004. See Table 2, column (4.8^2).
%H A249565 I. Jensen, and A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/31/40/008">Self-avoiding walks, neighbour-avoiding walks and trails on semi-regular lattices</a>, J. Phys. A., 31, (1998), 8137-45.
%H A249565 Keh Ying Lin and Chi Chen Chang, <a href="https://doi.org/10.1142/S0217979202010117">Self-avoiding walks on the 4-8 lattice</a>, International Journal of Modern Physics B, 16 (2002), 1241-1246.
%H A249565 Wikipedia, <a href="http://en.wikipedia.org/wiki/Truncated_square_tiling">Truncated square tiling</a>
%H A249565 Wikipedia, <a href="http://en.wikipedia.org/wiki/Connective_constant">Connective constant</a>
%H A249565 M. Zabrocki, <a href="http://garsia.math.yorku.ca/fieldseminar/notes141031.pdf">SAWs and SAPs on the Cayley graph of a group</a>, notes 2014.
%e A249565 There are 6 paths of length 2 in the truncated square lattice corresponding to the reduced words in the Coxeter group s_0 s_2, s_0 s_1, s_1 s_2, s_1 s_0, s_2 s_0, s_2 s_1.
%Y A249565 Cf. A001411, A001334.
%K A249565 nonn,walk
%O A249565 0,2
%A A249565 _Mike Zabrocki_, Nov 01 2014
%E A249565 a(20)-a(21) from _Mike Zabrocki_, Nov 08 2014
%E A249565 a(19)-a(21) corrected based on Alm (2005) and Lin & Chang (2002), more terms added by _Andrey Zabolotskiy_, Oct 18 2024