1,2

A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in E.Weissteins MathWorld article is shown in "Asymptotic Behavior of Mean Square Displacement" at first link

See under A001411

I. Jensen, Table of n, a(n) for n = 1..59 [from the Jensen link below]

I. Jensen, Series Expansions for Self-Avoiding Walks

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk

Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant, Section from World of Mathematics

a(n)=sum k=1, A046661(n) ( i_k^2 + j_k^2 ) where (i_k, j_k) are the end points of all different self-avoiding n-step walks

Example: a(2)=8 because the A046661(2)=3 different self-avoiding 2-step walks end at (1,-1),(1,1)->d^2=2 and at (2,0)->d^2=4, so a(2) = 2*2 + 1*4 = 8 a(3)=41 because the end-points of the 9 different 3-step walks are: (0,-1),(0,1)->d^2=1, (1,-2),(1,2),(2,-1),(2,-1),(2,1),(2,1)->d^2=5, (3,0)->d^2=9. a(3) = 2*1 + 6*5 + 1*9 = 41 See also "Distribution of end point distance" at first link

Source code of "FORTRAN program for distance counting" available at first link

Cf. A001411, A046661.

Sequence in context: A041116 A135797 A133106 this_sequence A156790 A080840 A026968

Adjacent sequences: A078794 A078795 A078796 this_sequence A078798 A078799 A078800

frac,nonn

Hugo Pfoertner (hugo(AT)pfoertner.org), Dec 05 2002