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A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.
(Formerly M4197 N1751)

%I M4197 N1751

%S 1,6,30,138,618,2730,11946,51882,224130,964134,4133166,17668938,

%T 75355206,320734686,1362791250,5781765582,24497330322,103673967882,

%U 438296739594,1851231376374,7812439620678,32944292555934,138825972053046

%N Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.

%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

%D A. J. Guttmann, Asymptotic analysis of power-series expansions, pp. 1-234 of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989.

%D B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 459.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H I. Jensen, <a href="/A001334/b001334.txt">Table of n, a(n) for n = 0..40</a> (from the Jensen link below)

%H Sergio Caracciolo, Anthony J. Guttmann, Iwan Jensen, Andrea Pelissetto, Andrew N. Rogers, Alan D. Sokal, <a href="http://dx.doi.org/10.1007/s10955-005-7004-3">Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks</a>, Journal of Statistical Physics, September 2005, Volume 120, Issue 5, pp. 1037-1100.

%H M. E. Fisher and M. F. Sykes, <a href="http://dx.doi.org/10.1103/PhysRev.114.45">Excluded-volume problem and the Ising model of ferromagnetism</a>, Phys. Rev. 114 (1959), 45-58.

%H A. J. Guttmann, <a href="/A002906/a002906.pdf">Asymptotic analysis of power-series expansions</a>, pp. 1-13, 56-57, 142-143, 150-151 from of C. Domb and J. L. Lebowitz, editors, Phase Transitions and Critical Phenomena. Vol. 13, Academic Press, NY, 1989. (Annotated scanned copy)

%H A. J. Guttmann and J. Wang, <a href="https://doi.org/10.1088/0305-4470/24/13/024">The extension of self-avoiding random walk series in two dimensions</a>, J. Phys. A 24 (1991), 3107-3109.

%H B. D. Hughes, Random Walks and Random Environments, vol. 1, Oxford 1995, <a href="/A001334/a001334.pdf">Tables and references for self-avoiding walks counts</a> [Annotated scanned copy of several pages of a preprint or a draft of chapter 7 "The self-avoiding walk"]

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

%H J. L. Martin, M. F. Sykes and F. T. Hioe, <a href="http://dx.doi.org/10.1063/1.1841242">Probability of initial ring closure for self-avoiding walks on the face-centered cubic and triangular lattices</a>, J. Chem. Phys., 46 (1967), 3478-3481.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>

%H D. C. Rapaport, <a href="https://doi.org/10.1088/0305-4470/18/4/003">End-to-end distance of linear polymers in two dimensions: a reassessment</a>, J. Phys. A 18 (1985), L201.

%H S. Redner, <a href="http://physics.bu.edu/~redner/pubs/pdf/jpa13p3525.pdf">Distribution functions in the interior of polymer chains</a>, J. Phys. A 13 (1980), 3525-3541, doi:10.1088/0305-4470/13/11/023.

%H M. F. Sykes, <a href="http://dx.doi.org/10.1063/1.1724212">Some counting theorems in the theory of the Ising problem and the excluded volume problem</a>, J. Math. Phys., 2 (1961), 52-62.

%H Joris van der Hoeven, <a href="http://dx.doi.org/10.1016/j.jsc.2009.01.001">On asymptotic extrapolation</a>, Journal of symbolic computation, 2009, p. 1010.

%t mo={{2, 0},{-1, 1},{-1, -1},{-2, 0},{1, -1},{1, 1}}; a[0]=1;

%t a[tg_, p_:{{0, 0}}] := Block[{e, mv = Complement[Last[p]+# & /@ mo, p]}, If[tg == 1, Length@mv, Sum[a[tg-1, Append[p, e]], {e, mv}]]];

%t a /@ Range[0, 6]

%t (* _Robert FERREOL_, Nov 28 2018; after the program of Giovanni Resta in A001411 *)

%o (Python)

%o def add(L,x):

%o ... M=[y for y in L];M.append(x)

%o ... return(M)

%o plus=lambda L,M : [x+y for x,y in zip(L,M)]

%o mo=[[2,0],[-1,1],[-1, -1],[-2,0],[1,-1],[1, 1]]

%o def a(n,P=[[0, 0]]):

%o ... if n==0: return(1)

%o ... mv1 = [plus(P[-1],x) for x in mo]

%o ... mv2=[x for x in mv1 if x not in P]

%o ... if n==1: return(len(mv2))

%o ... else: return(sum(a(n-1,add(P,x)) for x in mv2))

%o [a(n) for n in range(11)]

%o # _Robert FERREOL_, Dec 11 2018

%Y Cf. A001411, A001412, A001336, A001666, A001335, A036418, A192208.

%K nonn,walk,nice

%O 0,2

%A _N. J. A. Sloane_

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Last modified May 19 16:51 EDT 2019. Contains 323395 sequences. (Running on oeis4.)