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%I #7 Apr 26 2021 15:00:43
%S 6,23,80,263,834,2569,7764,23095,67910,197607,570560,1635331,4661026,
%T 13212739,37296004,104836893,293710714,820132581,2283926980,
%U 6343214871,17578257134,48604029143,134141458280,369519394643
%N 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).
%C 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.
%D See under A001411.
%H Hugo Pfoertner, <a href="http://www.randomwalk.de/stw2d.html">Results for the 2D Self-Trapping Random Walk</a>
%F 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.
%e 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.
%o Source code of "FORTRAN program for distance counting" available at first link
%Y Cf. A001411, A046661, A078797.
%K frac,nonn
%O 2,1
%A _Hugo Pfoertner_, Dec 10 2002
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