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

1, 3, 6, 12, 22, 42, 80, 152, 284, 536, 988, 1848, 3412, 6352, 11724, 21718, 39952, 73808, 135668, 250188, 459172, 844888, 1548608, 2845186, 5211548, 9563768, 17501272, 32079524, 58660712, 107425356, 196320596, 359232144, 656099656, 1199676412, 2189995764

Offset: 0

Author

Mike Zabrocki, Nov 01 2014

Keywords

• nonn
• walk

Comments

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.

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).

Examples

			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.
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 0..47 (from Alm, 2005)
• Sven Erick Alm, Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices, J. Phys. A.: Math. Gen., 38 (2005), 2055-2080. Also technical report of the same name, 2004. See Table 2, column (4.8^2).
• I. Jensen, and A. J. Guttmann, Self-avoiding walks, neighbour-avoiding walks and trails on semi-regular lattices, J. Phys. A., 31, (1998), 8137-45.
• Keh Ying Lin and Chi Chen Chang, Self-avoiding walks on the 4-8 lattice, International Journal of Modern Physics B, 16 (2002), 1241-1246.
• Wikipedia, Truncated square tiling
• Wikipedia, Connective constant
• M. Zabrocki, SAWs and SAPs on the Cayley graph of a group, notes 2014.

Crossrefs

Cf. A001411, A001334.

Extensions

a(20)-a(21) from Mike Zabrocki, Nov 08 2014

a(19)-a(21) corrected based on Alm (2005) and Lin & Chang (2002), more terms added by Andrey Zabolotskiy, Oct 18 2024