A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.

10, 67, 396, 2201, 11870, 62571, 324896, 1665349, 8457890, 42605267, 213305636, 1061939193, 5263752278, 25984214383, 127848694424, 627084275649, 3067923454498

Offset

2,1

Comments

A conjectured asymptotic behavior for the mean Manhattan displacement is shown in a diagram lim n-> infinity a(n)/(A078717(n)*n^nu)=c, for some values of nu near 0.59 at Pfoertner link

References

See under A001412

Links

Table of n, a(n) for n=2..18.

Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk

Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant

Formula

a(n)= sum l=1, A078717(n) (|i_l| + |j_l| + |k_l|) where (i_l, j_l, k_l) are the end points of all different self-avoiding n-step walks starting at (0, 0, 0)

Example

a(2)=10 because the A078717(2)=5 different self-avoiding 2-step walks end at (1,0,-1),(1,0,1),(1,-1,0),(1,1,0),(2,0,0)->d=2. a(2)=5*2=10. See also "Distribution of end point distance" at Pfoertner link

Prog

(Fortran) c Program for distance counting available at Pfoertner link.

Crossrefs

Cf. A001412, A078717, A078605 (corresponding square displacement).

Sequence in context: A026875 A026868 A026886 * A324366 A197783 A197751

Adjacent sequences: A079153 A079154 A079155 * A079157 A079158 A079159

Keyword

more,nonn

Author

Hugo Pfoertner, Dec 29 2002

Status

approved