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A001334 Number of n-step self-avoiding walks on hexagonal [ =triangular ] lattice.
(Formerly M4197 N1751)
1, 6, 30, 138, 618, 2730, 11946, 51882, 224130, 964134, 4133166, 17668938, 75355206, 320734686, 1362791250, 5781765582, 24497330322, 103673967882, 438296739594, 1851231376374, 7812439620678, 32944292555934, 138825972053046 (list; graph; refs; listen; history; text; internal format)



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


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).


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.


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 *)



def add(L, x):

    M=[y for y in L]; M.append(x)


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


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

Sequence in context: A002920 A255463 A192208 * A125316 A092439 A082149

Adjacent sequences:  A001331 A001332 A001333 * A001335 A001336 A001337




N. J. A. Sloane



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Last modified May 15 10:54 EDT 2021. Contains 343909 sequences. (Running on oeis4.)