A342800 Number of self-avoiding polygons on a 3-dimensional cubic lattice where each walk consists of steps with incrementing length from 1 to n.
0, 0, 0, 0, 0, 0, 24, 72, 0, 0, 1704, 5184, 0, 0, 193344, 600504, 0, 0, 34321512, 141520752, 0, 0, 9205815672, 37962945288, 0, 0
Offset: 1
Examples
a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
a(7) = 24 as there is one walk which forms a closed loop which can be walked in 24 different ways on a 3D cubic lattice. These walks, and those for n(8) = 72, are purely 2-dimensional. See A334720 for images of these walks.
a(11) = 1704. These walks consist of 120 purely 2-dimensional walks and 1584 3-dimensional walks. One of these 3-dimensional walks is:
.
/|
/ | z y
/ | | /
7 +y / | |/
/ | 8 -z |----- x
6 +x / |
|---.---.---.---.---.---/ | 9 +x
| |---.---.---.---.---.---.---.---.---/
| 5 +z /
| /
|---.---.---.---/ /
4 -x / 3 +y /
/ / 10 -y
| 2 +z /
| /
| 1 +z /
X---.---.---.---.---.---.---.---.---.---.---/
11 -x
.
Links
- A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
Comments