A352838 -id:A352838 - 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

A307468 Cogrowth sequence for the Heisenberg group.

Original entry on oeis.org

1, 4, 28, 232, 2156, 21944, 240280, 2787320, 33820044, 424925872, 5486681368, 72398776344, 972270849512, 13247921422480, 182729003683352, 2546778437385032, 35816909974343308, 507700854900783784, 7246857513425470288, 104083322583897779656

Offset: 0

Igor Pak, Apr 09 2019

This is the number of words of length 2n in the letters x,x^{-1},y,y^{-1} that equal the identity of the Heisenberg group H=.

Also, this is the number of closed walks of length 2n on the square lattice enclosing algebraic area 0 [Béguin et al.]. - Andrey Zabolotskiy, Sep 15 2021

			For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.

Jay Pantone, Table of n, a(n) for n = 0..200
Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
D. Lind and K, Schmidt, A survey of algebraic actions of the discrete Heisenberg group, arXiv:1502.06243 [math.DS], 2015; Russian Mathematical Surveys, 70:4 (2015), 77-142.

Related cogrowth sequences: Z A000984, Z^2 A002894, Z^3 A002896, (Z/kZ)^*2 for k = 2..5: A126869, A047098, A107026, A304979, Richard Thompson's group F A246877. The cogrowth sequences for BS(N,N) for N = 2..10 are A229644, A229645, A229646, A229647, A229648, A229649, A229650, A229651, A229652.

Cf. A352838, A178106.

Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).

A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.

Original entry on oeis.org

1, 6, 0, 6, 66, 0, 12, 0, 150, 0, 30, 1020, 0, 420, 0, 84, 0, 6, 0, 3444, 0, 1302, 0, 252, 0, 42, 19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24, 0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18, 449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30

Offset: 0

Andrey Zabolotskiy, Apr 22 2022

Rows 0 and 2 have 1 element each; row 1 is empty; for n > 2, we have 0 <= k <= A069813(n).

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002898.

			The triangle begins:
[1]
[]
[6]
[0, 6]
[66, 0, 12]
[0, 150, 0, 30]
[1020, 0, 420, 0, 84, 0, 6]
[0, 3444, 0, 1302, 0, 252, 0, 42]
[19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24]
[0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18]
[449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30]
...

Cf. A069813 (greatest area), A002898 (all closed walks), A352838 (square lattice).

For n > 1, row n seems to end with A109047(n).

A385672 Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 4, 12, 2, 40, 8, 4, 124, 42, 16, 6, 2, 416, 160, 92, 28, 16, 4, 4, 1348, 678, 362, 174, 88, 34, 22, 8, 6, 2, 4624, 2548, 1624, 756, 460, 200, 156, 56, 40, 20, 12, 4, 4, 15632, 10062, 6336, 3586, 2110, 1106, 742, 388, 278, 152, 82, 46, 34, 14, 8, 6, 2

Offset: 0

Andrei Zabolotskii, Aug 04 2025

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give 4^n.

The algebraic area is Integral y dx over the walk, which equals (Sum_{steps right} y) - (Sum_{steps left} y).

			The triangle begins:
     1
     4
    12,   2
    40,   8,   4
   124,  42,  16,   6,  2
   416, 160,  92,  28, 16,  4,  4
  1348, 678, 362, 174, 88, 34, 22, 8, 6, 2
   ...
T(3, 1) = 8: RUR (right, up, right), LUR, RDL, LDL, URU, URD, DLU, DLD.

Row lengths are A033638 = A002620 + 1.
A352838 is an analog that gives the number of closed walks.
Cf. A029552, A098613.

d = [{((0, 0), 0): 1}]
for _ in range(10):
    nd = {}
    for key, nw in d[-1].items():
        pos, ar = key
        x, y = pos
        for key in [
            ((x+1, y), ar + y),
            ((x-1, y), ar - y),
            ((x, y+1), ar),
            ((x, y-1), ar)
            ]:
            if key in nd:
                nd[key] += nw
            else:
                nd[key] = nw
    d.append(nd)
t = []
for nd in d:
    a = [0] * (max(ar for _, ar in nd) + 1)
    for key, nw in nd.items():
        _, ar = key
        if ar >= 0:
            a[ar] += nw
    t.append(a)
print(t)

It appears that T(2*n, n^2 - k) = 2 * A029552(k) for k < n and T(2*n+1, n^2+n - k) = 4 * A098613(k) for k < n.

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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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A307468 Cogrowth sequence for the Heisenberg group.

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A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.

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A385672 Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

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