A358046 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 only visited lattice points are considered when determining the visibility of points.

4, 8, 32, 64, 240, 480, 1904, 3832, 13992, 29304, 103088, 219416, 765600, 1609176, 5611680, 11785240, 40641032, 86254960, 293015872, 628547128, 2108574592, 4556118936, 15143701888, 32875906992, 108521571624, 236390241280, 776007097296, 1695412485136, 5538287862344

Offset: 1

Author

Scott R. Shannon, Oct 26 2022

Keywords

• nonn
• walk

Comments

Consider a self-avoiding walk on a 2D square lattice where two visited lattice points are considered to be visible from each other if either no other lattice points exist on the line drawn directly between these two lattice points, or if such points exist, they have not been visited by previous steps 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.

For the walks studied there is a difference in the ratio for the number of end-to-end visible walks to all walks for steps with even-n to odd-n. For example a(28)/A001411(28) ~ 0.72, while a(29)/A001411(29) ~ 0.88. The values and behavior of these ratios as n -> infinity is unknown.

See A358036 for the number of walks where the path between lattice points is also considered when determining point visibility.

Examples

			a(1) = 4 as after one step in any of the four available directions the lattice point stepped to and the starting point have no other points between them, so the first point is visible from the last for all four walks.
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.
a(3) = 32 as there are thirty-six 3-step SAWs, and of those, only the four walks directly along the axes have visited points between the first and last points, so a(3) = 36 - 4 = 32.
a(4) = 64 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 four other walks which have points on the line between the first and last point, and these points have been visited by earlier steps. These walks are:
.
     X            .---X          X
     |            |              |
     @---.        @          @---.       .---.
         |        |          |           |   |
     X---.    X---.      X---.       X---@   X
.
where the first and last points are shown as 'X' and where the visited points on the line between these two 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 - 4*8 = 100 - 36 = 64.
		

Links

• A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
• A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Crossrefs

Cf. A358036 (include path), A001411, A334877, A347990, A347506, A337353, A336262.