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A358036 Number of n-step self-avoiding walks on a 2D square lattice where the first visited lattice point is directly visible from the last visited lattice point, and were both the visited lattice points and the path between these points are considered when determining the visibility of points.

%I A358036 #21 Oct 30 2022 23:04:12
%S A358036 0,8,24,48,144,336,992,2344,6760,16336,46432,113904,320864,793136,
%T A358036 2222824,5524040,15409704,38493560,106895408,268253720,742053704,
%U A358036 1869175480,5154271008,13022699248,35816428904,90722285632,248960813992,631978627880,1730939615552
%N A358036 Number of n-step self-avoiding walks on a 2D square lattice where the first visited lattice point is directly visible from the last visited lattice point, and were both the visited lattice points and the path between these points are considered when determining the visibility of points.
%C A358036 Consider a self-avoiding walk on a 2D square lattice where two visited lattice points are considered to be visible from each other if, on drawing a line directly between these two points, the line neither crosses another lattice point which has been visited by previous steps of the walk, nor crosses any line directly connecting two consecutively visited lattice points that forms a part of the path of the walk. This sequence lists the number of n-step self-avoiding walks for which the first visited lattice point of the walk is directly visible from the last visited point. See the examples below.
%C A358036 For the 29-step walk the ratio of the number of end-to-end visible walks to all walks is a(29)/A001411(29) = 1730939615552/6279396229332 ~ 0.276. The value and behavior of this ratio as n -> infinity is unknown.
%C A358036 See A358046 for the number of walks when only the visited lattice points are considered when determining point visibility.
%H A358036 A. R. Conway et al., <a href="http://dx.doi.org/10.1088/0305-4470/26/7/012">Algebraic techniques for enumerating self-avoiding walks on the square lattice</a>, J. Phys A 26 (1993) 1519-1534.
%H A358036 A. J. Guttmann and A. R. Conway, <a href="http://dx.doi.org/10.1007/PL00013842">Self-Avoiding Walks and Polygons</a>, Annals of Combinatorics 5 (2001) 319-345.
%e A358036 a(1) = 0 as after one step in any of the four available directions the first and last point of the walk are directly connected by a line forming the path, so are not considered mutually visible.
%e A358036 a(2) = 8 as there are 4*3 = 12 2-step SAWs, but the four walks which consist of two steps directly along the axes have a visited lattice point directly between the first and last points of the walk, so those two point are not visible from each other. Thus a(2) = 12 - 4 = 8.
%e A358036 a(3) = 24 as there are thirty-six 3-step SAWs which include four walks directly along the axes which have a first point that is not visible from the last. In the first quadrant there is one other walk whose second-step path is intersected by the line between the first and last points of the walk. This walk is:
%e A358036 .
%e A358036        .---X
%e A358036        |
%e A358036    X---.
%e A358036 .
%e A358036 where the first and last points are shown as 'X'. The above walk can be walked in eight ways on the 2D square lattice, so the total number of walks where the first point is visible from the last is 36 - 4 - 1*8 = 36 - 12 = 24.
%e A358036 a(4) = 48 as there are one hundred 4-step SAWs which include four walks directly along the axes which have a first point that is not visible from the last. In the first quadrant there are six other walks which have either previously visited points directly on the line between the first and last points of the walk, or in which this line intersects the path of previous steps. These walks are:
%e A358036 .
%e A358036    X           .---X        X
%e A358036    |           |            |
%e A358036    @---.       @        @---.      .---.---X     .---.           .---X
%e A358036        |       |        |          |             |   |           |
%e A358036    X---.   X---.    X---.      X---.         X---@   X   X---.---.
%e A358036 .
%e A358036 where the visited points on the line between the first and last points are shown as '@'. Each of the above walks can be walked in eight ways on the 2D square lattice, so the total number of walks where the first point is visible from the last is 100 - 4 - 6*8 = 100 - 52 = 48.
%Y A358036 Cf. A358046 (ignore path), A001411, A334877, A347990, A347506, A337353, A336262.
%K A358036 nonn,walk
%O A358036 1,2
%A A358036 _Scott R. Shannon_, Oct 26 2022