A078797 Sum of square displacements over all self-avoiding n-step walks on a square lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/A046661(n).

1, 8, 41, 176, 679, 2452, 8447, 28120, 91147, 289324, 902721, 2777112, 8441319, 25398500, 75744301, 224156984, 658855781, 1924932324, 5593580859, 16175728584, 46572304083, 133556779740, 381611332725, 1086759598120

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

Cf. A001411, A046661, A176177.

Sequence in context: A358588 A272843 A268997 * A176177 A368529 A273112

Adjacent sequences: A078794 A078795 A078796 * A078798 A078799 A078800

Keyword

frac,nonn

Author

Hugo Pfoertner, Dec 05 2002

Extensions

Name amended by Scott R. Shannon, Sep 15 2020

Status

approved