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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.

D. C. Rapaport, J. Phys. A 18 (1985), L201.

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.

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

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.

Unknown author, Tables of self-avoiding walks, [Annotated scanned copy. Unfortunately the author and title of this document have been mislaid]

Joris van der Hoeven, On asymptotic extrapolation, Journal of symbolic computation, 2009, p. 1010.


Cf. 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 February 22 18:16 EST 2018. Contains 299469 sequences. (Running on oeis4.)