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

1,2

Comments

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

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.

Example

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.

Prog

FORTRAN program for distance counting available at Pfoertner link.

Crossrefs

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

Equals A118313/6.

Sequence in context: A027255 A121791 A016753 * A021029 A270496 A128594

Adjacent sequences: A078602 A078603 A078604 * A078606 A078607 A078608

Keyword

nonn

Author

Hugo Pfoertner, Dec 09 2002

Extensions

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

Name amended by Scott R. Shannon, Sep 17 2020

Status

approved