A010566 Number of 2n-step 2-dimensional closed self-avoiding paths on square lattice.

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• walk

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.

Crossrefs

Cf. A002931.

Programs

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 *)

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