%I #8 Mar 28 2021 00:19:57
%S 1,6,30,150,750,3750,18630,92406,458262,2270478,11245590,55697766,
%T 275769654,1365260862,6758345838,33450929886,165549052326,
%U 819248589606,4054005363918
%N Number of self-avoiding walks on a 3-dimensional cubic lattice where the walk consists of steps with incrementing length from 1 to n.
%C This sequence gives the number of self-avoiding walks on a 3-dimensional cubic lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. The first time a collision with a previous step can occur is for n = 6. See A334877 for further details.
%H A. J. Guttmann and A. R. Conway, <a href="http://dx.doi.org/10.1007/PL00013842">Self-Avoiding Walks and Polygons</a>, Annals of Combinatorics 5 (2001) 319-345.
%e a(1) to a(5) = 6*5^(n-1) as the number of walks equals the total number of non-backtracking walks when collisions are ignored.
%e a(6) = 18630 as, given one or more steps to the right followed by an upward step, the total number of walks that collide with a previous step is 5. These steps can be taking in 4*6 = 24 ways on the cubic lattice, giving 5*24 = 120 walks in all that are eliminated. The total number of walks ignoring collisions is 6*5^5 = 18750, so the total number of self-avoiding walks is 18750-120 = 18630.
%Y Cf. A334877, A342800, A001412, A334602, A336262.
%K nonn,more
%O 0,2
%A _Scott R. Shannon_, Mar 22 2021