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A078605
Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).
5
1, 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212, 64177763220925, 322314275563424, 1613192327878789, 8049191357609204, 40048773875769449, 198750753713937600
OFFSET
1,2
COMMENTS
A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in Weisstein's article is shown in "Asymptotic Behavior of Mean Square Displacement" at the Pfoertner link.
REFERENCES
For references see under A001412
LINKS
Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, Exact enumeration of self-avoiding walks, arXiv:1104.2184 [math-ph], 2011.
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant.
FORMULA
a(n) = Sum_{L=1..A001412(n)/6} ( i_L^2 + j_L^2 + k_L^2 ) where (i_L, j_L, k_L) are the endpoints of all different self-avoiding n-step walks.
EXAMPLE
a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.
PROG
(Fortran) c Program for distance counting available at Pfoertner link.
CROSSREFS
Cf. A001412, A078717, A079156 (corresponding Manhattan distance sum).
Equals A118313/6.
Sequence in context: A027255 A121791 A016753 * A021029 A270496 A128594
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Dec 09 2002
EXTENSIONS
Terms a(19)-a(36) taken from A118313 by Hugo Pfoertner, Aug 20 2014
Name amended by Scott R. Shannon, Sep 17 2020
STATUS
approved