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A010568 Number of 2n-step self-avoiding closed paths on the 4-dimensional cubic lattice.

%I A010568 #40 Jun 25 2025 09:41:50
%S A010568 8,48,912,22944,652320,20266368,669756192,23146172544,827460518688,
%T A010568 30378237727200,1139447186954208,43501453488658368,1685588678025512832
%N A010568 Number of 2n-step self-avoiding closed paths on the 4-dimensional cubic lattice.
%C A010568 From _M. F. Hasler_, Jun 18 2025: (Start)
%C A010568 This sequence gives the number of self-avoiding paths starting and ending at the origin. It does not consider equivalence with respect to translations, rotations or reflections. So it does count multiple cycles whose set of edges represents the same polygon. For example, a(2) = 48 counts 4-step cycles which all correspond to a unit square.
%C A010568 This explains why 8n | a(n) for all n, and a(n)/8n = (1, 3, 38, 717, 16308, 422216, 11959932, 361658946, 11492507204, 379727971590, ...)
%C A010568 Also, a(n = 1) = 8 gives the 2-step cycles which correspond to one step in any of the 8 directions, and one step back. Although this path is self-avoiding since it does not cross any point multiple times, it uses the same edge for the first and second step, and therefore might be excluded from the counting. In three dimensions this is done in A001413, A001413(1) = 0, as opposed to A010567(1) = 6. For two dimensions, this alternate definition is also used in A010566(1) = 0.
%C A010568 (End)
%H A010568 Nathan Clisby, Richard Liang, and Gordon Slade, <a href="https://doi.org/10.1088/1751-8113/40/36/003">Self-avoiding walk enumeration via the lace expansion</a>, J. Phys. A: Math. Theor. 40 (2007), 10973-11017. Table A6 "Enumeration results for d = 4", column p_n, row 2*n gives a(n)/(4*n) for n>1.
%H A010568 Nathan Clisby, Richard Liang, and Gordon Slade, <a href="https://personal.math.ubc.ca/~slade/lacecounts/">Self-avoiding walk enumeration via the lace expansion</a>. [Tables in machine-readable format on separate pages.]
%H A010568 M. E. Fisher and D. S. Gaunt, <a href="http://dx.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.
%F A010568 For all n, a(n) is divisible by 8*n. - _M. F. Hasler_, Jun 18 2025
%o A010568 (Python)
%o A010568 def A010568(n): # For illustration - becomes slow for n > 5
%o A010568     if not hasattr(A:=A010568, 'r'):
%o A010568        A.terms = [8]; O = 0,; I = O*4, (1,*O*3)
%o A010568        A.paths = (*I, (2,*O*3)), (*I, (1,1,0,0))
%o A010568     while n > len(A.terms):
%o A010568         for L in (0, 1):
%o A010568             new=[]; cycles = 0; O=(0,)*4; I = 0,1,2,3
%o A010568             for path in A.paths:
%o A010568                 end = path[-1]; weight = 48 if path[2][1] else 8
%o A010568                 for i in I:
%o A010568                    for s in (1, -1):
%o A010568                       t = tuple(end[j]if j!=i else end[j]+s for j in I)
%o A010568                       if t not in path: new.append(path+(t,))
%o A010568                       elif L and t==O: cycles += weight
%o A010568             A.paths = new
%o A010568         A.terms.append(cycles)
%o A010568     return A.terms[n-1] # _M. F. Hasler_, Jun 17 2025
%Y A010568 Cf. A010566, A010567, A010569, A010570 (similar for dimension 2 through 6).
%K A010568 nonn,walk,more
%O A010568 1,1
%A A010568 _N. J. A. Sloane_
%E A010568 a(6)-a(8) from _Sean A. Irvine_, May 31 2018
%E A010568 a(9)-a(10) from _Sean A. Irvine_, Aug 09 2020
%E A010568 "Self-avoiding" inserted in definition by _M. F. Hasler_, Jun 18 2025
%E A010568 a(11)-a(13) from Clisby et al.'s data added by _Andrei Zabolotskii_, Jun 25 2025