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

Author

Scott R. Shannon, Mar 21 2021

Keywords

• nonn
• more

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.

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.

Crossrefs

Cf. A334720, A334877, A342807, A001413, A002896, A002899, A010566, A335305.