A078798 Sum of Manhattan distances over all self-avoiding n-step walks on square lattice. Numerator of mean Manhattan displacement s(n) = a(n)/A046661(n).

6, 23, 80, 263, 834, 2569, 7764, 23095, 67910, 197607, 570560, 1635331, 4661026, 13212739, 37296004, 104836893, 293710714, 820132581, 2283926980, 6343214871, 17578257134, 48604029143, 134141458280, 369519394643

Offset

2,1

Comments

A conjectured asymptotic behavior for the mean Manhattan displacement lim n-> infinity a(n)/(A046661(n)*n^(3/4)) = constant is illustrated in "Asymptotic Behavior of Mean Manhattan Displacement" at first link.

References

See under A001411.

Links

Table of n, a(n) for n=2..25.

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk

Formula

a(n) = sum k=1, A046661(n) (|i_k| + |j_k|) where (i_k, j_k) are the end points of all different self-avoiding n-step walks.

Example

a(3)=23 because 2 of the A046661(3)=9 walks end at Manhattan distance 1: (0,-1),(0,1) and 7 walks end at Manhattan distance 3: (1,-2),(1,2),2*(2,-1),2*(2,1),(3,0); a(3)=2*1+7*3=23 See also "Distribution of end point distance" at first link.

Prog

Source code of "FORTRAN program for distance counting" available at first link

Crossrefs

Cf. A001411, A046661, A078797.

Sequence in context: A220148 A114245 A220238 * A027043 A006815 A264690

Adjacent sequences: A078795 A078796 A078797 * A078799 A078800 A078801

Keyword

frac,nonn

Author

Hugo Pfoertner, Dec 10 2002

Status

approved