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

Author

Scott R. Shannon, Sep 28 2020

Keywords

• nonn
• more
• walk

Comments

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.

Examples

			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.
		

Crossrefs

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