|
| |
|
|
A078797
|
|
Sum of square displacements over all self-avoiding n-step walks on square lattice. Numerator of mean square displacement s(n)=a(n)/A046661(n).
|
|
4
| |
|
|
1, 8, 41, 176, 679, 2452, 8447, 28120, 91147, 289324, 902721, 2777112, 8441319, 25398500, 75744301, 224156984, 658855781, 1924932324, 5593580859, 16175728584, 46572304083, 133556779740, 381611332725, 1086759598120
(list; graph; refs; listen; history; internal format)
|
|
|
|
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 E.Weissteins MathWorld article is shown in "Asymptotic Behavior of Mean Square Displacement" at first link
|
|
|
REFERENCES
| See under A001411
|
|
|
LINKS
| I. Jensen, Table of n, a(n) for n = 1..59 [from the Jensen link below]
I. Jensen, Series Expansions for Self-Avoiding Walks
Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant, Section from World of Mathematics
|
|
|
FORMULA
| a(n)=sum k=1, A046661(n) ( i_k^2 + j_k^2 ) where (i_k, j_k) are the end points of all different self-avoiding n-step walks
|
|
|
EXAMPLE
| Example: a(2)=8 because the A046661(2)=3 different self-avoiding 2-step walks end at (1,-1),(1,1)->d^2=2 and at (2,0)->d^2=4, so a(2) = 2*2 + 1*4 = 8 a(3)=41 because the end-points of the 9 different 3-step walks are: (0,-1),(0,1)->d^2=1, (1,-2),(1,2),(2,-1),(2,-1),(2,1),(2,1)->d^2=5, (3,0)->d^2=9. a(3) = 2*1 + 6*5 + 1*9 = 41 See also "Distribution of end point distance" at first link
|
|
|
PROG
| Source code of "FORTRAN program for distance counting" available at first link
|
|
|
CROSSREFS
| Cf. A001411, A046661.
Sequence in context: A135797 A171714 A133106 * A176177 A156790 A080840
Adjacent sequences: A078794 A078795 A078796 * A078798 A078799 A078800
|
|
|
KEYWORD
| frac,nonn
|
|
|
AUTHOR
| Hugo Pfoertner (hugo(AT)pfoertner.org), Dec 05 2002
|
| |
|
|
