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A001334
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Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.
(Formerly M4197 N1751)
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23
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1, 6, 30, 138, 618, 2730, 11946, 51882, 224130, 964134, 4133166, 17668938, 75355206, 320734686, 1362791250, 5781765582, 24497330322, 103673967882, 438296739594, 1851231376374, 7812439620678, 32944292555934, 138825972053046
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OFFSET
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0,2
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COMMENTS
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The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
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REFERENCES
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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).
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LINKS
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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.
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)
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MATHEMATICA
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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]
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PROG
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(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)]
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CROSSREFS
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KEYWORD
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nonn,walk,nice
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AUTHOR
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STATUS
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approved
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