A010569
Number of 2n-step self-avoiding closed paths on the 5-dimensional cubic lattice.
Original entry on oeis.org
10, 80, 2160, 82720, 3737120, 186303840, 9945915840, 558476528000, 32597366872320, 1961752814181280, 121020530395783040, 7620016712806580160
Offset: 1
- Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor. 40 (2007), 10973-11017. Table A7 "Enumeration results for d = 5", column p_n, row 2*n gives a(n)/(4*n) for n>1.
- Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]
- 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.
-
def A010569(n): # For illustration - becomes slow for n >= 5
if not hasattr(A:=A010569, 'r'):
A.terms = [10]; A.r = 0,1,2,3,4; z = 0,0,0,0; I = (0,*z), (1,*z)
A.paths = (*I,(2,*z)), (*I,(1,1,*z[1:])); A.weights = 10, 80
while n > len(A.terms):
for L in (0, 1):
np = []; nw=[];cycles = 0
for path,weight in zip(A.paths,A.weights):
end = path[-1]
for i in A.r:
for s in (1, -1):
t = tuple(end[j]if j!=i else end[j]+s for j in A.r)
if t not in path: np+=[path+(t,)]; nw+=[weight]
elif L and t==path[0]: cycles += weight
A.paths, A.weights = np, nw
A.terms.append(cycles)
return A.terms[n-1] # M. F. Hasler, Jun 17 2025
"Self-avoiding" inserted in definition by
M. F. Hasler, Jun 18 2025
A010567
Number of 2n-step self-avoiding closed paths (or cycles) on the 3-dimensional cubic lattice.
Original entry on oeis.org
6, 24, 264, 3312, 48240, 762096, 12673920, 218904768, 3891176352, 70742410800, 1309643747808, 24609869536800, 468270744898944, 9005391024862848, 174776445357365040, 3419171337633496704
Offset: 1
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.
-
def A010567(n): # For illustration - becomes slow for n > 5
if not hasattr(A:=A010567, 'terms'):
A.terms=[6]; O=0,; A.paths=[(O*3, (1,)+O*2, t+O)for t in((2,0),(1,1))]
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 += 24 if path[2][1] else 6
A.paths = new
A.terms.append(cycles)
return A.terms[n-1] # M. F. Hasler, Jun 17 2025
Name edited and "self-avoiding" added by
M. F. Hasler, Jun 17 2025
A010568
Number of 2n-step self-avoiding closed paths on the 4-dimensional cubic lattice.
Original entry on oeis.org
8, 48, 912, 22944, 652320, 20266368, 669756192, 23146172544, 827460518688, 30378237727200, 1139447186954208, 43501453488658368, 1685588678025512832
Offset: 1
- Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion, 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.
- Nathan Clisby, Richard Liang, and Gordon Slade, Self-avoiding walk enumeration via the lace expansion. [Tables in machine-readable format on separate pages.]
- 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.
-
def A010568(n): # For illustration - becomes slow for n > 5
if not hasattr(A:=A010568, 'r'):
A.terms = [8]; O = 0,; I = O*4, (1,*O*3)
A.paths = (*I, (2,*O*3)), (*I, (1,1,0,0))
while n > len(A.terms):
for L in (0, 1):
new=[]; cycles = 0; O=(0,)*4; I = 0,1,2,3
for path in A.paths:
end = path[-1]; weight = 48 if path[2][1] else 8
for i in I:
for s in (1, -1):
t = tuple(end[j]if j!=i else end[j]+s for j in I)
if t not in path: new.append(path+(t,))
elif L and t==O: cycles += weight
A.paths = new
A.terms.append(cycles)
return A.terms[n-1] # M. F. Hasler, Jun 17 2025
"Self-avoiding" inserted in definition by
M. F. Hasler, Jun 18 2025
Showing 1-3 of 3 results.
Comments