A079158 Sum of end-to-end Manhattan distances over all self-avoiding walks on cubic lattice trapped after n steps.

5, 40, 399, 2472, 17436, 98400, 601626, 3238694, 18355742, 96020478

Offset

11,1

Comments

Mean Manhattan displacement is a(n)/A077817(n).

See also "Comparison of average Euclidean and Manhattan displacements" at link

Links

Table of n, a(n) for n=11..20.

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

Formula

a(n)= Sum_{l=1..A077817(n)} (|i_l| + |j_l| + |k_l|) where (i_l, j_l, k_l) are the end points of all different self-avoiding walks trapped after n steps.

Example

a(12)=40 because the A077817(12)=20 trapped walks stop at 5*(1,1,0)->d=2, 5*(2,0,0)->d=2, 10*(1,0,1)->d=2, so a(12)=5*2+5*2+10*2=40. See "Enumeration of all self-trapping walks of length 12" at link.

Prog

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

Crossrefs

Cf. A077817, A079156, A079157 (corresponding squared distance sum).

Sequence in context: A368011 A124555 A152601 * A061633 A371372 A143437

Adjacent sequences: A079155 A079156 A079157 * A079159 A079160 A079161

Keyword

more,nonn

Author

Hugo Pfoertner, Dec 30 2002

Extensions

a(19)-a(20) from Sean A. Irvine, Jul 31 2025

Status

approved