OFFSET
1,2
COMMENTS
Original name: Number of 2n-step polygons on the cubic lattice. "Polygons" suggests translation invariance, and/or independence of the starting point; those are counted in A001409. Here, two paths which start and end at the origin are counted as distinct if they have a different sequence of steps, even if their set of edges is the same modulo translations.
a(n) is the number of 2n-step closed self-avoiding paths on the cubic lattice. - Bert Dobbelaere, Jan 04 2019
REFERENCES
B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 462.
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).
LINKS
M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
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.
M. F. Sykes, D. S. McKenzie, M. G. Watts, and J. L. Martin, The number of selfavoiding rings on a lattice, J. Phys. A 5 (1972), 661-666.
FORMULA
a(n) = 4*n*A001409(n). - Sean A. Irvine, Jul 27 2020
PROG
(Python)
def A001413(n): # For illustration; becomes slow for n >= 5.
if not hasattr(A:=A001413, 'terms'): A.terms=[]; A.paths=((0, 0, 0), ),
while n > len(A.terms):
for L in (0, 1):
new = []; cycles = 0
for path in A.paths:
end = path[-1]
for i in (0, 1, 2):
for s in (1, -1):
t = tuple(end[j]if j!=i else end[j]+s for j in (0, 1, 2))
if t not in path: new.append(path+(t, ))
elif L and t==path[0]: cycles += 1
A.paths = new
A.terms.append(cycles)
return A.terms[n-1] if n > 1 else 0 # M. F. Hasler, Jun 17 2025
CROSSREFS
KEYWORD
nonn,walk,more
AUTHOR
EXTENSIONS
a(11)-a(12) from Bert Dobbelaere, Jan 04 2019
a(13)-a(16) (using A001409) from Alois P. Heinz, Feb 28 2024
Name changed by M. F. Hasler, Jun 17 2025
STATUS
approved
