A342807 Number of self-avoiding walks on a 3-dimensional cubic lattice where the walk consists of steps with incrementing length from 1 to n.

1, 6, 30, 150, 750, 3750, 18630, 92406, 458262, 2270478, 11245590, 55697766, 275769654, 1365260862, 6758345838, 33450929886, 165549052326, 819248589606, 4054005363918

Offset: 0

Author

Scott R. Shannon, Mar 22 2021

Keywords

• nonn
• more

Comments

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.

Examples

			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.
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.
		

Links

• A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Crossrefs

Cf. A334877, A342800, A001412, A334602, A336262.