A008855 -id:A008855 - cp's OEIS frontend

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

Scott R. Shannon, May 10 2020

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.

			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.

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.

Cf. A008855, A010566, A002931, A316194, A001411, A037245, A352838.

T(n, k) = 4 * n * A008855(k, n). - Andrey Zabolotskiy, Sep 27 2024

A178106 Number of closed walks of length 16 and algebraic area n in the square lattice.

Original entry on oeis.org

33820044, 28133728, 18569808, 10127744, 5015108, 2289760, 1036368, 435040, 184104, 73056, 28064, 10336, 3760, 1088, 352, 96, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Jonathan Vos Post, Dec 16 2010

n=16 column of Table 1.1, p.9, Mohammad-Noori. Histogram of the number of closed walks for n = 16, 32, 64.

The algebraic area is a signed quantity, and can become zero for self-intersecting paths.

Morteza Mohammad-Noori, Enumeration of closed random walks in the square lattice according to their areas, arXiv:1012.3720, Dec 16, 2010.

Cf. A002894, A008855.

a(0) + 2*Sum_{n>0} a(n) = A002894(8).

A259857 Triangle T(n,k), n>=1, 2<=k<=n+1, read by rows, where T(n,k) is the number of self-avoiding square-lattice polygons by area n and perimeter 2*k.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 1, 18, 0, 0, 0, 8, 55, 0, 0, 0, 2, 40, 174, 0, 0, 0, 0, 22, 168, 566, 0, 0, 0, 0, 6, 134, 676, 1868, 0, 0, 0, 0, 1, 72, 656, 2672, 6237, 0, 0, 0, 0, 0, 30, 482, 2992, 10376, 21050, 0, 0, 0, 0, 0, 8, 310, 2592, 13160, 39824, 71666, 0, 0, 0, 0, 0, 2, 151, 2086, 12862, 56162, 151878, 245696

Offset: 1

N. J. A. Sloane, Jul 07 2015

			Triangle begins:
==========================================================
n\k  | 2 3 4  5  6   7   8    9    10    11     12     13
-----|----------------------------------------------------
   1 | 1,
   2 | 0,2,
   3 | 0,0,6,
   4 | 0,0,1,18
   5 | 0,0,0, 8,55,
   6 | 0,0,0, 2,40,174,
   7 | 0,0,0, 0,22,168,566,
   8 | 0,0,0, 0, 6,134,676,1868,
   9 | 0,0,0, 0, 1, 72,656,2672, 6237,
  10 | 0,0,0, 0, 0, 30,482,2992,10376,21050,
  11 | 0,0,0, 0, 0,  8,310,2592,13160,39824, 71666,
  12 | 0,0,0, 0, 0,  2,151,2086,12862,56162,151878,245696,

I. G. Enting and A. J. Guttmann, On the area of square lattice polygons, J. Statist. Phys., 58 (1990), 475-484. See Table 1.

A006725 and A006726 are diagonals.
Row sums give A006724.
Cf. A008855 (with 0 omitted).

a(7)-a(10) inserted by Seiichi Manyama, Apr 04 2020

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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.

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A178106 Number of closed walks of length 16 and algebraic area n in the square lattice.

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A259857 Triangle T(n,k), n>=1, 2<=k<=n+1, read by rows, where T(n,k) is the number of self-avoiding square-lattice polygons by area n and perimeter 2*k.

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