OFFSET
1,2
COMMENTS
A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in E. Weisstein's MathWorld article is shown in the "Asymptotic Behavior of Mean Square Displacement" link. [I'm not sure this comment is correct. There may be some confusion with A176177. - N. J. A. Sloane, Aug 02 2015]
REFERENCES
See under A001411
LINKS
I. Jensen, Table of n, a(n) for n = 1..59 [from the Jensen link below]
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
Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant
FORMULA
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
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
PROG
(FORTRAN) See Hugo Pfoertner link for source code of "FORTRAN program for distance counting".
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Hugo Pfoertner, Dec 05 2002
EXTENSIONS
Name amended by Scott R. Shannon, Sep 15 2020
STATUS
approved