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%I #19 Jul 25 2020 09:52:09
%S 0,0,8,32,152,512,1880,5920,19464,59168,183776,545392,1638400,4778000,
%T 14043224,40422544,116977176,333346928,953538440,2695689520,
%U 7642091352,21464794032,60417010152,168787016352,472315518008,1313548558528,3657850909680,10133559518800
%N Total number of neighbor contacts for n-step self-avoiding walks on a 2D square lattice.
%C This sequence gives the total number of neighbor contacts for all n-step self avoiding walks on a 2D square lattices. A neighbor contact is when the walk comes within 1 unit distance of a previously visited point, excluding the previous adjacent point.
%H D. Bennett-Wood, I. G. Enting, D. S. Gaunt, A. J. Guttmann, J. L. Leask, A. L. Owczarek, and S. G. Whittington, <a href="https://doi.org/10.1088/0305-4470/31/20/010">Exact enumeration study of free energies of interacting polygons and walks in two dimensions</a>, J. Phys. A: Math. Gen. 31 (1998), 4725-4741.
%H M. E. Fisher and B. J. Hiley, <a href="http://dx.doi.org/10.1063/1.1731729">Configuration and free energy of a polymer molecule with solvent interaction</a>, J. Chem. Phys., 34 (1961), 1253-1267.
%H A. M. Nemirovsky, K. F. Freed, T. Ishinabe, and J. F. Douglas, <a href="http://dx.doi.org/10.1007/BF01049010">Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers</a>, J. Statist. Phys., 67 (1992), 1083-1108.
%e a(1) = a(2) = 0 as a 1 and 2 step walk cannot approach a previous step.
%e a(3) = 8. The single walk where one interaction occurs, which can be taken in eight ways on a 2D square lattice, is:
%e .
%e +---+
%e |
%e X---+
%e .
%e Therefore, the total number of interactions is 1*1*8 = 8.
%e a(4) = 32. The four walks where one interaction occurs, each of which can be taken in eight ways on a 2D square lattice, are:
%e .
%e +---+---+ + +---+ +---+
%e | | | | |
%e X---+ +---+ X---+---+ X---+ +
%e |
%e X---+
%e .
%e Therefore, the total number of interactions is 4*1*8 = 32.
%e a(5) = 152. Considering only walks which start with one or more steps to the right followed by an upward step there are thirty-five different walks. Eleven of these have one neighbor contact (hence A033155(5) = 11*8 = 88) while four have two contacts. These are:
%e .
%e +---+---+ +---+---+ +---+ +---+
%e | | | | | |
%e + X---+ X---+---+ +---+ + +
%e | |
%e X---+ X---+
%e .
%e Therefore, the total number of contacts is (11*1 + 4*2)*8 = 152.
%Y Cf. A033155 (total number of n-step walks containing one neighbor contact), A038747, A047057, A173380, A174319.
%K nonn
%O 1,3
%A _Scott R. Shannon_, Jul 23 2020