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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.

%I A342800 #12 Mar 28 2021 00:19:47
%S A342800 0,0,0,0,0,0,24,72,0,0,1704,5184,0,0,193344,600504,0,0,34321512,
%T A342800 141520752,0,0,9205815672,37962945288,0,0
%N 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.
%C A342800 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.
%H A342800 A. J. Guttmann and A. R. Conway, <a href="http://dx.doi.org/10.1007/PL00013842">Self-Avoiding Walks and Polygons</a>, Annals of Combinatorics 5 (2001) 319-345.
%e A342800 a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
%e A342800 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.
%e A342800 a(11) = 1704. These walks consist of 120 purely 2-dimensional walks and 1584 3-dimensional walks. One of these 3-dimensional walks is:
%e A342800 .
%e A342800                                 /|
%e A342800                                / |                        z  y
%e A342800                               /  |                        | /
%e A342800                         7 +y /   |                        |/
%e A342800                             /    | 8 -z                   |----- x
%e A342800              6 +x          /     |
%e A342800   |---.---.---.---.---.---/      |               9 +x
%e A342800   |                              |---.---.---.---.---.---.---.---.---/
%e A342800   | 5 +z                                                            /
%e A342800   |                                                                /
%e A342800   |---.---.---.---/                                               /
%e A342800         4 -x     /  3 +y                                         /
%e A342800                 /                                               /  10 -y
%e A342800                 | 2 +z                                         /
%e A342800                 |                                             /
%e A342800                 | 1 +z                                       /
%e A342800                 X---.---.---.---.---.---.---.---.---.---.---/
%e A342800                                      11 -x
%e A342800 .
%Y A342800 Cf. A334720, A334877, A342807, A001413, A002896, A002899, A010566, A335305.
%K A342800 nonn,more
%O A342800 1,7
%A A342800 _Scott R. Shannon_, Mar 21 2021