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A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.

%I A079156 #14 Feb 16 2025 08:32:48
%S A079156 10,67,396,2201,11870,62571,324896,1665349,8457890,42605267,213305636,
%T A079156 1061939193,5263752278,25984214383,127848694424,627084275649,
%U A079156 3067923454498
%N A079156 Sum of end-to-end Manhattan distances over all self-avoiding n-step walks on cubic lattice. Numerator of mean Manhattan displacement s(n)=a(n)/A078717.
%C A079156 A conjectured asymptotic behavior for the mean Manhattan displacement is shown in a diagram lim n-> infinity a(n)/(A078717(n)*n^nu)=c, for some values of nu near 0.59 at Pfoertner link
%D A079156 See under A001412
%H A079156 Hugo Pfoertner, <a href="http://www.randomwalk.de/stw3d.html">Results for the 3-dimensional Self-Trapping Random Walk</a>
%H A079156 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Self-AvoidingWalkConnectiveConstant.html">Self-Avoiding Walk Connective Constant</a>
%F A079156 a(n)= sum l=1, A078717(n) (|i_l| + |j_l| + |k_l|) where (i_l, j_l, k_l) are the end points of all different self-avoiding n-step walks starting at (0, 0, 0)
%e A079156 a(2)=10 because the A078717(2)=5 different self-avoiding 2-step walks end at (1,0,-1),(1,0,1),(1,-1,0),(1,1,0),(2,0,0)->d=2. a(2)=5*2=10. See also "Distribution of end point distance" at Pfoertner link
%o A079156 (Fortran) c Program for distance counting available at Pfoertner link.
%Y A079156 Cf. A001412, A078717, A078605 (corresponding square displacement).
%K A079156 more,nonn
%O A079156 2,1
%A A079156 _Hugo Pfoertner_, Dec 29 2002