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A078605 Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).

%I A078605 #40 Feb 16 2025 08:32:48
%S A078605 1,12,97,672,4261,25588,147821,830576,4566917,24692980,131682825,
%T A078605 694386864,3626770709,18790632772,96675376705,494382431552,
%U A078605 2514666026897,12730690730212,64177763220925,322314275563424,1613192327878789,8049191357609204,40048773875769449,198750753713937600
%N A078605 Sum of square displacements over all self-avoiding n-step walks on the cubic lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/(A001412(n)/6).
%C A078605 A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in Weisstein's article is shown in "Asymptotic Behavior of Mean Square Displacement" at the Pfoertner link.
%D A078605 For references see under A001412
%H A078605 Hugo Pfoertner, <a href="/A078605/b078605.txt">Table of n, a(n) for n = 1..36</a>
%H A078605 Hugo Pfoertner, <a href="https://www.randomwalk.de/stw3d.html">Results for the 3-dimensional Self-Trapping Random Walk</a>
%H A078605 Raoul D. Schram, Gerard T. Barkema, and Rob H. Bisseling, <a href="http://arxiv.org/abs/1104.2184">Exact enumeration of self-avoiding walks</a>, arXiv:1104.2184 [math-ph], 2011.
%H A078605 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Self-AvoidingWalkConnectiveConstant.html">Self-Avoiding Walk Connective Constant</a>.
%F A078605 a(n) = Sum_{L=1..A001412(n)/6} ( i_L^2 + j_L^2 + k_L^2 ) where (i_L, j_L, k_L) are the endpoints of all different self-avoiding n-step walks.
%e A078605 a(2)=12 because the A001412(2)/6 = 5 different self-avoiding 2-step walks end at (1,0,-1), (1,0,1), (1,-1,0), (1,1,0)->d^2=2 and at (2,0,0)->d^2=4. a(2) = 4*2 + 1*4 = 12. See also "Distribution of end point distance" at first link.
%o A078605 (Fortran) c Program for distance counting available at Pfoertner link.
%Y A078605 Cf. A001412, A078717, A079156 (corresponding Manhattan distance sum).
%Y A078605 Equals A118313/6.
%K A078605 nonn
%O A078605 1,2
%A A078605 _Hugo Pfoertner_, Dec 09 2002
%E A078605 Terms a(19)-a(36) taken from A118313 by _Hugo Pfoertner_, Aug 20 2014
%E A078605 Name amended by _Scott R. Shannon_, Sep 17 2020