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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.

This is a variation of A337860 where only walks which are stable against a small perturbation from either left or right are counted. This means any walks which have their center-of-mass directly above the extrema of the nodes touching the y=0 starting line are not counted, e.g. a walk directly up from the first node.

See A337860 for further details and examples of the walks in this sequence.

Consider a self-avoiding walk in the upper half-plane on a 2D square lattice where each visited node is given a fixed mass and each node is connected by a rod of the same mass. Let the resulting lattice structure be free to move in a downward gravitational field. This sequence gives the number of walks of length n such that the structure will remain in place and will not topple given no sideways perturbations.

For a walk to be stable requires the center-of-mass of the resulting structure to be above or inside the extrema of the horizontal positions of the nodes that are on the y=0 line where the walk begins. Here we assume no perturbations so allow walks which would topple if either a left or right perturbation acts, for example we allow a directly vertical walk above the starting node. For the number of walks where such semi-stable structures are not counted see A337317.

We also assume the nodes and the rods are of equal mass. This is required as some structures exist which are either stable or would topple depending on the relative mass of the nodes and rods. For example the 8-step walk:

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Considering only the nodes the center-of-mass is at position 17/9 (~1.88) relative to the starting x=0 'X' position - this is between the x=0 and x=2 extrema of the nodes at y=0 and is thus stable. Considering only the rods the center-of-mass is at position 33/16 (~2.06) relative to 'X' - this is to the right of the node at x=2 and thus the structure would topple to the right. To avoid such issues we assume both rods and nodes are of equal mass. Given that, the center-of-mass of this walk is at 67/34 (~1.97) and is thus stable.

The number of stable walks in this sequence does not decrease as rapidly as compared to the number of hanging 2D stable walks of A335780. For example the total number of 2D self-avoiding walks on a square lattice in the upper half plane for n=29 is A116903(27) = 1577923781445. The total number of vertically stable walks here for n=29 is 282209330237, indicating about 1 in 6 walks are stable. This is expected as many otherwise unstable walks becomes stable if some node touches the y=0 line away from the starting node; this becomes relatively common as n increases. Any of the symmetrical walks in A335780 which have no nodes above the starting node will also be in this sequence, inverted from top to bottom.

This is a variation of A337860 where at every step, given the nodes and connecting rods have equal mass, the resulting 2D lattice structure is stable against toppling, assuming no sideways perturbations. See that sequence for further details of the allowed walks.