A337860 -id:A337860 - cp's OEIS frontend

A337317 The number of stable vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

2, 4, 10, 24, 60, 138, 348, 832, 2104, 5192, 13178, 32662, 82890, 207888, 529738, 1339188, 3424526, 8698382, 22294906, 56836056, 145982928, 373363770, 960834764, 2463930512, 6351046936, 16322104184, 42131167144, 108478565772, 280360764620

Offset: 1

Scott R. Shannon, Sep 28 2020

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.

			a(1) = 2. The two stable walks are a single step left or right from the first node. The walk consisting of a single vertical step is not counted, as it has its center-of-mass directly above the single node touching the y=0 line and will thus topple with a slight perturbation from either the left or right directions.
a(3) = 10. The stable 3-step walks with a first step up or to the right are:
.
                                            +
+---+                         +  +---+      |
|   |  X---+---+---+          |      |      +
X   +                 X---+---+  X---+      |
                                        X---+
.
These walks can also be taken with a first or second step to the left, giving a total number of stable walks of 2*5 = 10.
The semi-stable 3-step walks which are not counted in this sequence, but are counted in A337860, are:
.
                        +
                        |
    +---+   +---+       +
    |           |       |
X---+           +---X   +
                        |
                        X
.
as a slight perturbation from the left, right, and left or right would topple the first, second and third structure respectively.

Cf. A337860 (count semi-stable walks), A335780, A337761, A116903, A116904, A001411.

A335098 The number of constructible vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

Original entry on oeis.org

3, 5, 11, 23, 51, 109, 251, 549, 1291, 2981, 7067, 16571, 39601, 94195, 226997, 544687, 1320935, 3194399, 7797891, 18996977, 46651387, 114353905, 282109663, 694793903, 1720327219, 4253521985, 10565387267, 26213565665, 65300013637, 162516950805, 405892537979

Offset: 1

Scott R. Shannon, Sep 12 2020

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.

			a(1) = 3, a(2) = 5. These are the same stable walks as in A337860.
a(3) = 11. The constructible stable walks given a first step to the right are:
.
                                                   +
                        +      +---+   +---+       |
                        |      |           |       +
X---+---+---+   X---+---+  X---+       X---+       |
                                               X---+
.
These walks can also take a first step to the left thus, along with the directly vertical walk, the total number of stable walks is 2*5 + 1 = 11.
One 3-step walk which is not counted here, along with its parent 2-step walk, is:
.
+---+        +---+
|      ==>   |   |
X            X   +
.
After two steps the resulting structure is not stable against toppling, its center-of-mass is clearly to the right of the one node at y=0, thus any resulting 3-step walks resulting from this unstable 2-step walk are not counted.

Cf. A337860, A337317, A335780, A337761, A116903, A116904, A001411.

cp's OEIS Frontend

A337317 The number of stable vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

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