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

(Fortran) c 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