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Search: id:A078798
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| A078798 |
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Sum of Manhattan distances over all self-avoiding n-step walks on square lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A046661(n). |
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2
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| 6, 23, 80, 263, 834, 2569, 7764, 23095, 67910, 197607, 570560, 1635331, 4661026, 13212739, 37296004, 104836893, 293710714, 820132581, 2283926980, 6343214871, 17578257134, 48604029143, 134141458280, 369519394643
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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A conjectured asymptotic behavior for the mean Manhattan displacement lim n-> infinity a(n)/(A046661(n)*n^(3/4))=constant is illustrated in "Asymptotic Behavior of Mean Manhattan Displacement" at first link
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REFERENCES
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See under A001411
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LINKS
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Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk
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FORMULA
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a(n)= sum k=1, A046661(n) (|i_k| + |j_k|) where (i_k, j_k) are the end points of all different self-avoiding n-step walks.
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EXAMPLE
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a(3)=23 because 2 of the A046661(3)=9 walks end at Manhattan distance 1: (0,-1),(0,1) and 7 walks end at Manhattan distance 3: (1,-2),(1,2),2*(2,-1),2*(2,1),(3,0); a(3)=2*1+7*3=23 See also "Distribution of end point distance" at first link
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PROGRAM
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Source code of "FORTRAN program for distance counting" available at first link
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CROSSREFS
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Cf. A001411, A046661, A078797.
Sequence in context: A058751 A034359 A114245 this_sequence A027043 A006815 A054491
Adjacent sequences: A078795 A078796 A078797 this_sequence A078799 A078800 A078801
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KEYWORD
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frac,nonn
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AUTHOR
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Hugo Pfoertner (hugo(AT)pfoertner.org), Dec 10 2002
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