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The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the connecting rods have mass.
1

%I #13 Oct 14 2020 23:15:49

%S 1,1,1,1,3,7,17,43,91,183,371,799,1941,4621,11463,27823,68997,167481,

%T 414045,1006091,2496981,6127053,15304071,37838777,95041475,236320611,

%U 595206771

%N The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the connecting rods have mass.

%C This is a variation of A335780 where only the rods between the nodes have mass. See that sequence for further details of the allowed walks.

%e a(1)-a(4) = 1 as the only stable walk is a walk straight down from the first node.

%e a(5) = 3. There is one stable walk with a first step to the right:

%e .

%e X-----+

%e |

%e |

%e +-----+-----+-----+

%e ,

%e Assuming a rod mass of q, the total torque to the right of the first node is 2*q*(1/2) + 1*q*1 = 2q. The total torque to the left of the first node is 1*q*(1/2) + 1*q*(3/2) = 2q. This walk can be taken in 2 ways. Thus, with the straight down walk, the total number of stable walks is 2+1 = 3.

%Y Cf. A335780 (rods and nodes have mass), A335307 (only nodes have mass), A116903, A337761, A001411, A001412.

%K nonn,walk,more

%O 1,5

%A _Scott R. Shannon_, Sep 13 2020