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After constructing a self-avoiding walk, bridge together all adjacent neighboring sites on the walk. This sequence is sum of the total number of links after adding bridges across all walks of length n. - Sean A. Irvine, Aug 09 2017

A self-avoiding walk is a sequence of adjacent points in a lattice that are all distinct. The truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. The edge lattice is also referred to as (4,8^2) lattice. It is also the Cayley graph of the Coxeter group generated by three generators {s_0, s_1, s_2} with the relations s_i^2 = 1, s_0 s_2 = s_2 s_0, (s_i s_{i+1})^4 = 1 for i=0,1.

It is conjectured that a(n) is approximately mu^n*n^{11/32} for large n where mu is the connective constant and mu is approximately 1.80883001(6).

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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X---+---+

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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.

The walks counted are all those directly along and x or y axes, and all walks whose final (|x|,|y|) lattice point are the two legs of a Pythagorean triple.

This is a cleaner representation than the one given by A077483 and A077484, using the upper bound for the denominator A077484 given in A076874.

In a 2D self-avoiding walk there may be steps, where the number of free target positions is less than 3. A step is called k-constrained, if only k<3 neighbors were not visited before. Self-trapping occurs at step n (the next step would have k=0). A maximally 2-constrained n-step walk contains n-floor((4*n+1)^(1/2))-2 steps with k=2 (conjectured). The first step is chosen fixed (0,0)->(1,0), all other steps have k=3. This sequence counts those walks among all possible self-trapping n-step walks A077482(n).