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A010566 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice.
(Formerly N1903)
5
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)
OFFSET

1,2

COMMENTS

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

REFERENCES

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.

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

B. Hayes, How to avoid yourself, American Scientist, Vol. 86, Number 4, Jul-August 1998. p. 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.

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

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

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

LINKS

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))

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

Iwan Jensen, Series Expansions for Self-Avoiding Walks

CROSSREFS

Sequence in context: A063515 A220706 A246030 * A305224 A182068 A092771

Adjacent sequences:  A010563 A010564 A010565 * A010567 A010568 A010569

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 14 09:49 EST 2018. Contains 317182 sequences. (Running on oeis4.)