A078605 Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).

1, 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212, 64177763220925, 322314275563424, 1613192327878789, 8049191357609204, 40048773875769449, 198750753713937600

Offset: 1

Author

Hugo Pfoertner, Dec 09 2002

Keywords

• 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 Weisstein's article is shown in "Asymptotic Behavior of Mean Square Displacement" at the Pfoertner link.

Examples

			a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.
		

References

• For references see under A001412

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..36
• Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk
• Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, Exact enumeration of self-avoiding walks, arXiv:1104.2184 [math-ph], 2011.
• Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant.

Crossrefs

Cf. A001412, A078717, A079156 (corresponding Manhattan distance sum).

Equals A118313/6.

Programs

Fortran

c Program for distance counting available at Pfoertner link.

Formula

a(n) = Sum_{L=1..A001412(n)/6} ( i_L^2 + j_L^2 + k_L^2 ) where (i_L, j_L, k_L) are the endpoints of all different self-avoiding n-step walks.

Extensions

Terms a(19)-a(36) taken from A118313 by Hugo Pfoertner, Aug 20 2014

Name amended by Scott R. Shannon, Sep 17 2020