A336448 Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.

0, 4, 32, 164, 704, 2716, 9808, 33788, 112480, 364588, 1157296, 3610884, 11108448, 33765276, 101594000, 302977204, 896627936, 2635423124, 7699729296, 22374323436, 64702914336, 186289216332, 534227118960, 1526445330900, 4347038392480, 12341626847324, 34940293640400, 98660244502668

Offset

0,2

Comments

See A001411 for the corresponding number of n-step self-avoiding walks.

References

See A001411 and A078797.

Links

Table of n, a(n) for n=0..27.

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

a(n) = Sum_{k=0..A001411(n)} ( i_k^2 + j_k^2 ) where (i_k, j_k) are the end points of all different self-avoiding n-step walks.

a(n) = 4*A078797(n).

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

Cf. A001411, A078797, A046661.

Sequence in context: A270619 A326745 A083299 * A259854 A272424 A269811

Adjacent sequences: A336445 A336446 A336447 * A336449 A336450 A336451

Keyword

nonn

Author

Scott R. Shannon, Jul 22 2020

Status

approved