OFFSET
0,2
COMMENTS
See A001411 for the corresponding number of n-step self-avoiding walks.
LINKS
A. J. Guttmann, On the critical behavior of self-avoiding walks, J. Phys. A 20 (1987), 1839-1854.
I. Jensen, Series Expansions for Self-Avoiding Walks
FORMULA
EXAMPLE
a(1) = 4 as a single step of length 1 can be taken in four ways on the square lattice the sum of square end-to-end displacements is 4*1 = 4.
a(2) = 32. The two 2-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
+
| 2 +---+---+ 4
+---+
.
The first walk can be taken in 8 ways on a square lattice, the latter in 4 ways, thus the total displacement over all 2-step walks is 8*2 + 4*4 = 32.
a(3) = 164. The five 3-step self-avoiding walks with a first step to the right in the first quadrant with their corresponding square displacements are:
.
+
+---+ | +---+ +
| 1 + 5 | 5 | 5 +---+---+---+ 9
+---+ | +---+ +---+---+
+---+
.
The first four walks can be taken in 8 ways on a square lattice, the last in 4 ways, thus the total displacement over all 3-step walks is 8*1 + 8*5 + 8*5 + 8*5 + 4*9 = 164.
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 22 2020
STATUS
approved