A038516
Total number of different legs traversed by all loops of length 2n in A038515.
Original entry on oeis.org
0, 0, 0, 25, 37, 151, 199, 457, 565, 1023, 1215, 1929, 2229
Offset: 0
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
A001667
2n-step polygons on b.c.c. lattice.
Original entry on oeis.org
96, 1776, 43776, 1237920, 37903776, 1223681760, 41040797376, 1416762272736, 50027402384640, 1799035070369856
Offset: 2
- 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).
- P. Butera and M. Comi, Enumeration of the self-avoiding polygons on a lattice by the Schwinger-Dyson equations, Annals of Combinatorics 3, 277-286 (1999); arXiv:cond-mat/9903297, 1999.
- M. E. Fisher and M. F. Sykes, Excluded-volume problem and the Ising model of ferromagnetism, Phys. Rev. 114 (1959), 45-58.
- M. F. Sykes et al., The number of self-avoiding walks on a lattice, J. Phys. A 5 (1972), 661-666.
- Index entries for sequences related to b.c.c. lattice
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