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A334756 Irregular table read by rows: T(n,k) is the number of 2n-step closed self-avoiding paths on a 2D square lattice with area k, where k >= n-1.

%I A334756 #34 Sep 27 2024 23:16:07
%S A334756 0,8,24,96,16,360,160,40,1320,960,528,144,24,4872,4704,3752,2016,840,
%T A334756 224,56,18112,21632,20992,15424,9920,4832,2176,704,192,32,67248,96192,
%U A334756 107712,93312,75096,50112,31104,16416,7848,3168,1080,288,72
%N A334756 Irregular table read by rows: T(n,k) is the number of 2n-step closed self-avoiding paths on a 2D square lattice with area k, where k >= n-1.
%C A334756 See A010566 for the number of closed self-avoiding 2D square lattice paths. Like that sequence here all possible paths are counted when determining the polygon areas, including those that are equivalent via rotation and reflection.
%H A334756 A. J. Guttmann and I. G. Enting, <a href="https://doi.org/10.1088/0305-4470/21/3/009">The size and number of rings on the square lattice</a>, J. Phys. A 21 (1988), L165-L172.
%H A334756 Brian Hayes, <a href="https://www.americanscientist.org/article/how-to-avoid-yourself">How to avoid yourself</a>, American Scientist 86 (1998) 314-319.
%H A334756 B. J. Hiley and M. F. Sykes, <a href="http://dx.doi.org/10.1063/1.1701041">Probability of initial ring closure in the restricted random-walk model of a macromolecule</a>, J. Chem. Phys., 34 (1961), 1531-1537.
%H A334756 Iwan Jensen, <a href="https://web.archive.org/web/20151222163324/http://www.ms.unimelb.edu.au/~iwan/saw/SAW_ser.html">Series Expansions for Self-Avoiding Walks</a>
%H A334756 G. S. Rushbrooke and J. Eve, <a href="http://dx.doi.org/10.1063/1.1730595">On Noncrossing Lattice Polygons</a>, Journal of Chemical Physics, 31 (1959), 1333-1334.
%H A334756 Scott R. Shannon, <a href="/A334756/a334756.dat.txt">Data for n=1..12</a>.
%F A334756 T(n, k) = 4 * n * A008855(k, n). - _Andrey Zabolotskiy_, Sep 27 2024
%e A334756 For n = 2, total steps = 4, there are 8 different paths with an area of 1. These are the 8 possible ways to walk the polygon:
%e A334756 +---+
%e A334756 |   |
%e A334756 +---+
%e A334756 .
%e A334756 For n = 3, total steps = 6, there are 24 different paths with an area of 2. These are the 24 possible ways to walk the polygon:
%e A334756 +---+---+
%e A334756 |       |
%e A334756 +---+---+
%e A334756 .
%e A334756 For n = 4, total steps = 8, there are 96 different paths with an area of 3 and 16 different paths with an area of 4. These are the possible ways to walk the polygons:
%e A334756 +---+                      +---+---+
%e A334756 |   |                      |       |
%e A334756 +   +---+                  +       +
%e A334756 |       |                  |       |
%e A334756 +---+---+  for area = 3    +---+---+ for area = 4
%e A334756 .
%e A334756 For n = 5, total steps = 10, there are 360 different paths with an area of 4, 160 paths with an area of 5 and 40 different paths with an area of 6. These are the possible ways to walk the polygons:
%e A334756 +---+---+---+---+    +---+               +---+           +---+---+
%e A334756 |               |    |   |               |   |           |       |
%e A334756 +---+---+---+---+    +   +---+---+   +---+   +---+   +---+   +---+
%e A334756                      |           |   |           |   |       |
%e A334756                      +---+---+---+   +---+---+---+   +---+---+      for area = 4
%e A334756 .
%e A334756 +---+---+                      +---+---+---+
%e A334756 |       |                      |           |
%e A334756 +       +---+                  +           +
%e A334756 |           |                  |           |
%e A334756 +---+---+---+  for area = 5    +---+---+---+  for area = 6
%e A334756 .
%e A334756 Table begins:
%e A334756 0;
%e A334756 8;
%e A334756 24;
%e A334756 96,16;
%e A334756 360,160,40;
%e A334756 1320,960,528,144,24;
%e A334756 4872,4704,3752,2016,840,224,56;
%e A334756 18112,21632,20992,15424,9920,4832,2176,704,192,32;
%e A334756 67248,96192,107712,93312,75096,50112,31104,16416,7848,3168,1080,288,72;
%e A334756 249480,415040,526400,514480,468680,373280,281280,189920,120400,69120,36560,17040,7480,2720,880,240,40;
%e A334756 Row sums = A010566.
%Y A334756 Cf. A008855, A010566, A002931, A316194, A001411, A037245, A352838.
%K A334756 nonn,tabf
%O A334756 1,2
%A A334756 _Scott R. Shannon_, May 10 2020