A343962 Number of self-avoiding walks that escape an n X n square lattice starting at a given corner.

4, 14, 106, 2142, 124150, 21231450, 10794801654, 16397345136778, 74754715306888786, 1026191624073867290710, 42506394853041064742716162, 5320474615969510569494723118086, 2014671515857822813610223858063766522

Offset

1,1

Comments

A self-avoiding walk on a square lattice allows horizontal and vertical movement one step at a time, where no space is visited more than once.

The n X n square can be seen as a subset of a larger lattice which surrounds it. Visiting any space on this larger lattice that is not part of the square constitutes escaping the square.

There are two ways to escape the square while standing at a corner, and both are counted separately.

a(n) is always even due to symmetry along a diagonal.

Links

Table of n, a(n) for n=1..13.

Example

For n=1, every direction will immediately result in escaping the board, so a(1) = 4.
For n=2, there are two ways to escape from the starting corner. Otherwise, any of the three remaining corners can be escaped from in two ways, and each corner can be reached from two different directions (clockwise and counterclockwise). Therefore a(2) = 2 + 3*2*2 = 14.

Crossrefs

Cf. A341269.

Sequence in context: A110302 A282617 A005743 * A283793 A048369 A269590

Adjacent sequences: A343959 A343960 A343961 * A343963 A343964 A343965

Keyword

nonn,walk,more

Author

Johan Westin, May 05 2021

Extensions

a(7)-a(13) from Andrew Howroyd, May 05 2021

Status

approved