A337761 The number of hanging vertically stable self-avoiding walks of length n on a 3D cubic lattice where both the nodes and connecting rods have mass.

1, 1, 1, 1, 1, 5, 13, 29, 73, 193, 581, 1717, 7029, 21981, 79625, 248697, 981353, 3275085, 13235333

Offset

1,6

Comments

Consider a self-avoiding walk on a 3D cubic lattice where each visited node is given a fixed mass and each node is connected by a rod of another fixed mass. Hang the resulting lattice structure from a string at the first node. This sequence gives the number of walks of length n such that the structure will hang perfectly vertically, and will return to this position if perturbed.

For a walk to be stable requires the torque around the first node to be zero along both the x and y axial directions for both the node and rod masses, and that the overall center of mass of the structure is lower than the first node. As n increases the number of walks satisfying these conditions decreases rapidly. For example the total number of 3D self-avoiding walks on a cubic lattice in the lower four quadrants for n=18 is A116904(18) = 719101961769. The total number of hanging stable walks for n=18 is 3275085, indicating only one in about 220 thousand walks is stable.

The stable walks are the same as in the 2D case, see A335780, up until n=9; the same stable single-plane walks occur but in both the x-z and y-z plane so the total of these walks is twice A335780. From n=10 stable walks can occur which use all three dimensions. For all stable walks it is found that the final node is always directly underneath the starting node. This is not the case if only the node or rod masses are considered.

See A335780 for the equalivalent sequence on a 2D square lattice.

Links

Table of n, a(n) for n=1..19.

Scott R. Shannon, Stable walks for n=6 to n=12.

Example

a(1)-a(5) = 1 as the only stable walk is a walk straight down from the first node.
a(6) = 5. There is one stable walk confined to a single plane:
.
      X-----+
            |
            |
+-----+-----+
|
|
+-----+
.
where 'X' represents the hanging point first node at (0,0,0).
Assuming a mass of p for the nodes, q for the rods, and a length l for the rods, the total torque from the nodes to the right of the first node is 2*p*l, which equals that from the nodes to the left. The total torque for the rods to the right of the first node is 2*q*(1/2)*l + 1*q*1*l = 2ql, which equals that from the rods to the left. The center of mass is at coordinate (0,0,-1). This walk can be taken in 4 ways thus, with the straight down walk, the total number of stable walks is 4+1 = 5.
a(10) = 193. Other than the straight and single plane walks those using three dimensions now occur. An example of such a walk is:
.
        +--------+      z    y
       /        /        \  /
      /   X-------+       \/
     /        /    \       +-----x
    /        /      \
   +        +        +
    \      /        /
     \    /        /
      +--/----+   /
        /        /
       +--------+
.
The total rotational torque around the y-axis from the nodes with x>0 is 3*p*l, which equals that from the nodes with x<0. The total rotational torque around the x-axis from the nodes with y>0 is 2*p*l, which equals that from the nodes with y<0. The total rotational torque around the y-axis from the rods with x>0 is 2*q*(1/2)*l + 2*q*1*l = 3ql, which equals that from the rods with x<0.  The total rotational torque around the x-axis from the rods with y>0 is 2*q*(1/2)*l + 1*q*1*l = 2ql, which equals that from the rods with x<0. The center of mass is at coordinate (0,0,-1).
a(11) = 581. An example of an 11 step stable walk where the final node is above the first node:
.
                   +              +
                  /              /|       z
                 /              / |       |  y
                /              /  |       | /
               +              /   |       |/
        +      |   X---------+    |       +-----x
       /|      |       +----------+
      / |      |      /
     /  |      |     /
    /   |      |    /
   +-----------+   /
        +---------+
.
This particular walk would be counted twice as it is also stable if hung from the final node.
See the linked text file for the step directions for the stable walks for n=6 to n=12.

Crossrefs

Cf. A335780, A001412, A116904, A001411.

Sequence in context: A295112 A239367 A124698 * A147278 A268963 A147349

Adjacent sequences: A337758 A337759 A337760 * A337762 A337763 A337764

Keyword

nonn,more,walk

Author

Scott R. Shannon, Sep 19 2020

Status

approved