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

Author

Hugo Pfoertner, Dec 05 2002

Keywords

• frac
• nonn

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]

Examples

			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
		

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

Crossrefs

Cf. A001411, A046661, A176177.

Programs

FORTRAN

See Hugo Pfoertner link for source code of "FORTRAN program for distance counting".

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.

Extensions

Name amended by Scott R. Shannon, Sep 15 2020