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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. 4
10, 67, 396, 2201, 11870, 62571, 324896, 1665349, 8457890, 42605267, 213305636, 1061939193, 5263752278, 25984214383, 127848694424, 627084275649, 3067923454498 (list; graph; refs; listen; history; text; internal format)
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 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

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Last modified August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)