A079157 Sum of square displacements over all self-avoiding walks on cubic lattice trapped after n steps. Numerator of mean square displacement a(n)/A077817(n).

5, 50, 529, 3870, 28900, 185014, 1191698, 7080332, 42072344, 238337862

Offset: 11

Author

Hugo Pfoertner, Dec 30 2002

Keywords

• more
• nonn

Examples

			a(12)=50 because the A077817(12)=20 trapped walks stop at 5*(1,1,0)->d^2=2, 5*(2,0,0)->d^2=4, 10*(1,0,1)->d^2=2. So, a(12)=5*2+5*4+10*2=50. See "Enumeration of all self-trapping walks of length 12" at link.
		

Links

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

Crossrefs

Cf. A077817, A078605, A079158 (corresponding Manhattan distance sum).

Programs

Fortran

c Program for distance counting available at link.

Formula

a(n) = Sum_{l=1..A077817(n)} (i_l^2 + j_l^2 + k_l^2) where (i_l, j_l, k_l) are the end points of all different self-avoiding walks trapped after n steps.

Extensions

a(15) corrected and a(19)-a(20) from Sean A. Irvine, Jul 31 2025