A078797 - OEIS

%I #27 Sep 16 2020 02:23:53

%S 1,8,41,176,679,2452,8447,28120,91147,289324,902721,2777112,8441319,

%T 25398500,75744301,224156984,658855781,1924932324,5593580859,

%U 16175728584,46572304083,133556779740,381611332725,1086759598120

%N Sum of square displacements over all self-avoiding n-step walks on a square lattice with the first step specified. Numerator of mean square displacement s(n)=a(n)/A046661(n).

%C A comparison with the conjectured asymptotic behavior of the mean square displacement s(n) over all n-step self-avoiding walks given in E. Weisstein's MathWorld article is shown in the "Asymptotic Behavior of Mean Square Displacement" link. [I'm not sure this comment is correct. There may be some confusion with A176177. - _N. J. A. Sloane_, Aug 02 2015]

%D See under A001411

%H I. Jensen, <a href="/A078797/b078797.txt">Table of n, a(n) for n = 1..59</a> [from the Jensen link below]

%H A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/20/7/029">On the critical behavior of self-avoiding walks</a>, J. Phys. A 20 (1987), 1839-1854.

%H I. Jensen, <a href="https://web.archive.org/web/20151222163324/http://www.ms.unimelb.edu.au/~iwan/saw/SAW_ser.html">Series Expansions for Self-Avoiding Walks</a>

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/stw2d.html">Results for the 2D Self-Trapping Random Walk</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Self-AvoidingWalkConnectiveConstant.html">Self-Avoiding Walk Connective Constant</a>

%F a(n) = Sum_{k=1..A046661(n)} ( i_k^2 + j_k^2 ) where (i_k, j_k) are the end points of all different self-avoiding n-step walks.

%e Example: a(2)=8 because the A046661(2)=3 different self-avoiding 2-step walks end at (1,-1),(1,1)->d^2=2 and at (2,0)->d^2=4, so a(2) = 2*2 + 1*4 = 8 a(3)=41 because the end-points of the 9 different 3-step walks are: (0,-1),(0,1)->d^2=1, (1,-2),(1,2),(2,-1),(2,-1),(2,1),(2,1)->d^2=5, (3,0)->d^2=9. a(3) = 2*1 + 6*5 + 1*9 = 41 See also "Distribution of end point distance" at first link

%o (FORTRAN) See Hugo Pfoertner link for source code of "FORTRAN program for distance counting".

%Y Cf. A001411, A046661, A176177.

%K frac,nonn

%O 1,2

%A _Hugo Pfoertner_, Dec 05 2002

%E Name amended by _Scott R. Shannon_, Sep 15 2020