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;

0, 8, 24, 96, 16, 360, 160, 40, 1320, 960, 528, 144, 24, 4872, 4704, 3752, 2016, 840, 224, 56, 18112, 21632, 20992, 15424, 9920, 4832, 2176, 704, 192, 32, 67248, 96192, 107712, 93312, 75096, 50112, 31104, 16416, 7848, 3168, 1080, 288, 72

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..43.

A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.

Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.

B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.

Iwan Jensen, Series Expansions for Self-Avoiding Walks

G. S. Rushbrooke and J. Eve, On Noncrossing Lattice Polygons, Journal of Chemical Physics, 31 (1959), 1333-1334.

Scott R. Shannon, Data for n=1..12.

Example

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:
+---+
|   |
+---+
.
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:
+---+---+
|       |
+---+---+
.
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:
+---+                      +---+---+
|   |                      |       |
+   +---+                  +       |
|       |                  |       |
+---+---+  for area = 3    +---+---+ for area = 4
.
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:
+---+---+---+---+    +---+               +---+           +---+---+
|               |    |   |               |   |           |       |
+---+---+---+---+    +   +---+---+   +---+   +---+   +---+   +---+
                     |           |   |           |   |       |
                     +---+---+---+   +---+---+---+   +---+---+      for area = 4
.
+---+---+                      +---+---+---+
|       |                      |           |
+       +---+                  +           +
|           |                  |           |
+---+---+---+  for area = 5    +---+---+---+  for area = 6
.
Table begins:
0;
8;
24;
96,16;
360,160,40;
1320,960,528,144,24;
4872,4704,3752,2016,840,224,56;
18112,21632,20992,15424,9920,4832,2176,704,192,32;
67248,96192,107712,93312,75096,50112,31104,16416,7848,3168,1080,288,72;
249480,415040,526400,514480,468680,373280,281280,189920,120400,69120,36560,17040,7480,2720,880,240,40;
Row sums = A010566.

Crossrefs

Cf A010566, A002931, A316194, A001411, A037245.

Sequence in context: A217016 A019260 A019221 * A281463 A267326 A063515

Adjacent sequences: A334753 A334754 A334755 * A334757 A334758 A334759

Keyword

nonn,tabf

Author

Scott R. Shannon, May 10 2020

Status

approved