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

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

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).

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))

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.

Mathematica

A002931 = Cases[Import["https://oeis.org/A002931/b002931.txt", "Table"], {_, _}][[All, 2]]; a[n_] := 4n A002931[[n]];
a /@ Range[55] (* Jean-François Alcover, Jan 11 2020 *)

Prog

(Python)
# Alex Nisnevich, Jul 22 2023
def num_continuations(path, dist):
    (x, y) = path[-1]
    next = [(x+1, y), (x-1, y), (x, y+1), (x, y-1)]
    if dist == 1:
        return (0, 0) in next
    else:
        return sum(num_continuations(path + [c], dist - 1) for c in next if c not in path)
def A010566(n):
    return 4 * num_continuations([(0, 0), (1, 0)], 2 * n - 1) if n >= 2 else 0

Crossrefs

Cf. A002931.

Sequence in context: A063515 A220706 A246030 * A305224 A182068 A092771

Adjacent sequences: A010563 A010564 A010565 * A010567 A010568 A010569

Keyword

nonn,nice,walk

Author

N. J. A. Sloane

Status

approved