cp's OEIS Frontend

A010567 Number of 2n-step self-avoiding closed paths (or cycles) on the 3-dimensional cubic lattice.

%I A010567 #29 Jun 19 2025 00:22:13
%S A010567 6,24,264,3312,48240,762096,12673920,218904768,3891176352,70742410800,
%T A010567 1309643747808,24609869536800,468270744898944,9005391024862848,
%U A010567 174776445357365040,3419171337633496704
%N A010567 Number of 2n-step self-avoiding closed paths (or cycles) on the 3-dimensional cubic lattice.
%C A010567 This sequence agrees with A001413 except for n=1, for which the given value is "purely conventional" (although the convention is non-standard): it counts 6 two-step closed paths, all of which visit no node twice but use an edge twice, so whether they are "self-avoiding" is indeed a matter of agreement. Same considerations apply to the first terms of A010568-A010570. - _Andrey Zabolotskiy_, May 29 2018
%H A010567 M. E. Fisher and D. S. Gaunt, <a href="https://doi.org/10.1103/PhysRev.133.A224">Ising model and self-avoiding walks on hypercubical lattices and high density expansions</a>, Phys. Rev. 133 (1964) A224-A239.
%o A010567 (Python)
%o A010567 def A010567(n): # For illustration - becomes slow for n > 5
%o A010567     if not hasattr(A:=A010567, 'terms'):
%o A010567         A.terms=[6]; O=0,; A.paths=[(O*3, (1,)+O*2, t+O)for t in((2,0),(1,1))]
%o A010567     while n > len(A.terms):
%o A010567         for L in (0,1):
%o A010567             new = []; cycles = 0
%o A010567             for path in A.paths:
%o A010567                 end = path[-1]
%o A010567                 for i in (0,1,2):
%o A010567                    for s in (1,-1):
%o A010567                       t = tuple(end[j]if j!=i else end[j]+s for j in (0,1,2))
%o A010567                       if t not in path: new.append(path+(t,))
%o A010567                       elif L and t==path[0]: cycles += 24 if path[2][1] else 6
%o A010567             A.paths = new
%o A010567         A.terms.append(cycles)
%o A010567     return A.terms[n-1] # _M. F. Hasler_, Jun 17 2025
%Y A010567 Essentially the same as A001413.
%Y A010567 Cf. A010568 (analog in 4 dimensions), A010569 (in 5D), A010570 (in 6D), A130706 (in 1D), A010566 (in 2D, different convention for n=1), A002896 (closed walks, not necessarily self-avoiding), A001412 (self-avoiding walks, not necessarily closed), A039618, A038515.
%K A010567 nonn,more
%O A010567 1,1
%A A010567 _N. J. A. Sloane_
%E A010567 a(8)-a(10) copied from A001413 by _Andrey Zabolotskiy_, May 29 2018
%E A010567 a(11)-a(12) copied from A001413 by _Pontus von Brömssen_, Feb 28 2024
%E A010567 a(13)-a(16) (using A001413) from _Alois P. Heinz_, Feb 28 2024
%E A010567 Name edited and "self-avoiding" added by _M. F. Hasler_, Jun 17 2025