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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.

%I A335098 #13 Oct 05 2020 12:15:17
%S A335098 3,5,11,23,51,109,251,549,1291,2981,7067,16571,39601,94195,226997,
%T A335098 544687,1320935,3194399,7797891,18996977,46651387,114353905,282109663,
%U A335098 694793903,1720327219,4253521985,10565387267,26213565665,65300013637,162516950805,405892537979
%N 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.
%C A335098 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.
%e A335098 a(1) = 3, a(2) = 5. These are the same stable walks as in A337860.
%e A335098 a(3) = 11. The constructible stable walks given a first step to the right are:
%e A335098 .
%e A335098                                                    +
%e A335098                         +      +---+   +---+       |
%e A335098                         |      |           |       +
%e A335098 X---+---+---+   X---+---+  X---+       X---+       |
%e A335098                                                X---+
%e A335098 .
%e A335098 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.
%e A335098 One 3-step walk which is not counted here, along with its parent 2-step walk, is:
%e A335098 .
%e A335098 +---+        +---+
%e A335098 |      ==>   |   |
%e A335098 X            X   +
%e A335098 .
%e A335098 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.
%Y A335098 Cf. A337860, A337317, A335780, A337761, A116903, A116904, A001411.
%K A335098 nonn,walk
%O A335098 1,1
%A A335098 _Scott R. Shannon_, Sep 12 2020