|
|
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.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
STATUS
|
approved
|
|
|
|