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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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1
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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
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OFFSET
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2,1
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COMMENTS
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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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LINKS
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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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PROG
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(Fortran) c Source code of "FORTRAN program for distance counting" available at first link.
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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