OFFSET
1,7
COMMENTS
This sequence gives the number of self-avoiding polygons (closed-loop self-avoiding walks) on a 3D cubic lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. Like A334720 and A335305 only n values corresponding to even triangular numbers can form closed loops. All possible paths are counted, including those that are equivalent via rotation and reflection.
LINKS
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
EXAMPLE
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
.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Scott R. Shannon, Mar 21 2021
STATUS
approved