|
Search: id:A078605
|
|
|
| A078605 |
|
Sum of square displacements over all self-avoiding n-step walks on cubic lattice. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6). |
|
+0
3
|
|
| 12, 97, 672, 4261, 25588, 147821, 830576, 4566917, 24692980, 131682825, 694386864, 3626770709, 18790632772, 96675376705, 494382431552, 2514666026897, 12730690730212
(list; graph; listen)
|
|
|
OFFSET
|
2,1
|
|
|
COMMENT
|
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
|
Hugo Pfoertner, Results for the 3-dimensional Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant, Section from World of Mathematics
|
|
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 end points 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
|
|
PROGRAM
|
FORTRAN program for distance counting available at Pfoertner link.
|
|
CROSSREFS
|
Cf. A001412, A078717, A079156 (corresponding Manhattan distance sum).
Sequence in context: A027255 A121791 A016753 this_sequence A021029 A128594 A166793
Adjacent sequences: A078602 A078603 A078604 this_sequence A078606 A078607 A078608
|
|
KEYWORD
|
more,nonn
|
|
AUTHOR
|
Hugo Pfoertner (hugo(AT)pfoertner.org), Dec 09 2002
|
|
|
Search completed in 0.002 seconds
|
