A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.
1, 6, 30, 138, 618, 2730, 11946, 51882, 224130, 964134, 4133166, 17668938, 75355206, 320734686, 1362791250, 5781765582, 24497330322, 103673967882, 438296739594, 1851231376374, 7812439620678, 32944292555934, 138825972053046
Offset: 0
References
- 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.
- B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 459.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- I. Jensen, Table of n, a(n) for n = 0..40 (from the Jensen link below)
- Sergio Caracciolo, Anthony J. Guttmann, Iwan Jensen, Andrea Pelissetto, Andrew N. Rogers, Alan D. Sokal, Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks, Journal of Statistical Physics, September 2005, Volume 120, Issue 5, pp. 1037-1100.
- M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
- A. J. Guttmann, Asymptotic analysis of power-series expansions, 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)
- A. J. Guttmann and J. Wang, The extension of self-avoiding random walk series in two dimensions, J. Phys. A 24 (1991), 3107-3109.
- B. D. Hughes, Random Walks and Random Environments, vol. 1, Oxford 1995, Tables and references for self-avoiding walks counts [Annotated scanned copy of several pages of a preprint or a draft of chapter 7 "The self-avoiding walk"]
- I. Jensen, Series Expansions for Self-Avoiding Walks
- J. L. Martin, M. F. Sykes and F. T. Hioe, Probability of initial ring closure for self-avoiding walks on the face-centered cubic and triangular lattices, J. Chem. Phys., 46 (1967), 3478-3481.
- G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
- D. C. Rapaport, End-to-end distance of linear polymers in two dimensions: a reassessment, J. Phys. A 18 (1985), L201.
- S. Redner, Distribution functions in the interior of polymer chains, J. Phys. A 13 (1980), 3525-3541, doi:10.1088/0305-4470/13/11/023.
- M. F. Sykes, Some counting theorems in the theory of the Ising problem and the excluded volume problem, J. Math. Phys., 2 (1961), 52-62.
- Joris van der Hoeven, On asymptotic extrapolation, Journal of symbolic computation, 2009, p. 1010.
Programs
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Mathematica
mo={{2, 0},{-1, 1},{-1, -1},{-2, 0},{1, -1},{1, 1}}; a[0]=1; 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}]]]; a /@ Range[0, 6] (* Robert FERREOL, Nov 28 2018; after the program of Giovanni Resta in A001411 *) -
Python
def add(L,x): M=[y for y in L];M.append(x) return(M) plus=lambda L,M : [x+y for x,y in zip(L,M)] mo=[[2,0],[-1,1],[-1, -1],[-2,0],[1,-1],[1, 1]] def a(n,P=[[0, 0]]): if n==0: return(1) mv1 = [plus(P[-1],x) for x in mo] mv2=[x for x in mv1 if x not in P] if n==1: return(len(mv2)) else: return(sum(a(n-1,add(P,x)) for x in mv2)) [a(n) for n in range(11)] # Robert FERREOL, Dec 11 2018
Comments