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A336448
Sum of square displacements over all n-step self-avoiding walks on a 2D square lattice.
3
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.
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
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 22 2020
STATUS
approved