%I #12 Mar 28 2021 00:19:47
%S 0,0,0,0,0,0,24,72,0,0,1704,5184,0,0,193344,600504,0,0,34321512,
%T 141520752,0,0,9205815672,37962945288,0,0
%N 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 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 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 a(1) to a(6) = 0 as no self-avoiding closed-loop walk is possible.
%e 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 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 .
%e /|
%e / | z y
%e / | | /
%e 7 +y / | |/
%e / | 8 -z |----- x
%e 6 +x / |
%e |---.---.---.---.---.---/ | 9 +x
%e | |---.---.---.---.---.---.---.---.---/
%e | 5 +z /
%e | /
%e |---.---.---.---/ /
%e 4 -x / 3 +y /
%e / / 10 -y
%e | 2 +z /
%e | /
%e | 1 +z /
%e X---.---.---.---.---.---.---.---.---.---.---/
%e 11 -x
%e .
%Y Cf. A334720, A334877, A342807, A001413, A002896, A002899, A010566, A335305.
%K nonn,more
%O 1,7
%A _Scott R. Shannon_, Mar 21 2021