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

Author

Scott R. Shannon, Jul 22 2020

Keywords

• nonn

Comments

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

Examples

			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.
		

References

• See A001411 and A078797.

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

Crossrefs

Cf. A001411, A078797, A046661.

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).