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).

1, 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: 1

Author

Hugo Pfoertner, Dec 10 2002

Keywords

• frac
• nonn

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.

Examples

			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.
		

References

• See under A001411.

Links

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

Crossrefs

Cf. A001411, A046661, A078797.

Programs

Fortran

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

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.

Extensions

a(1)=1 inserted by Sean A. Irvine, Jul 16 2025