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A010566 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice.
(Formerly N1903)
0, 8, 24, 112, 560, 2976, 16464, 94016, 549648, 3273040, 19781168, 121020960, 748039552, 4664263744, 29303071680, 185307690240, 1178635456752, 7535046744864, 48392012257184, 312061600211680, 2019822009608592, 13117263660884768, 85447982919036736 (list; graph; refs; listen; history; text; internal format)



a(n) = 4n*A002931(n). There are (2n) choices for the starting point and 2 choices for the orientation, in order to produce self-avoiding closed paths from a polygon of perimeter 2n. - Philippe Flajolet, Nov 22 2003


B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 461.

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


Felix A. Pahl, Table of n, a(n) for n = 1..55 (from Iwan Jensen's computations of A002931, using a(n)=4n*A002931(n))

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.

M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 364.

A. J. Guttmann and I. G. Enting, The size and number of rings on the square lattice, J. Phys. A 21 (1988), L165-L172.

Brian Hayes, How to avoid yourself, American Scientist 86 (1998) 314-319.

B. J. Hiley and M. F. Sykes, Probability of initial ring closure in the restricted random-walk model of a macromolecule, J. Chem. Phys., 34 (1961), 1531-1537.

Iwan Jensen, Series Expansions for Self-Avoiding Walks

G. S. Rushbrooke and J. Eve, On Noncrossing Lattice Polygons, Journal of Chemical Physics, 31 (1959), 1333-1334.


Cf. A002931.

Sequence in context: A063515 A220706 A246030 * A305224 A182068 A092771

Adjacent sequences:  A010563 A010564 A010565 * A010567 A010568 A010569




N. J. A. Sloane



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Last modified November 18 07:22 EST 2019. Contains 329252 sequences. (Running on oeis4.)