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

6, 24, 264, 3312, 48240, 762096, 12673920, 218904768, 3891176352, 70742410800, 1309643747808, 24609869536800, 468270744898944, 9005391024862848, 174776445357365040, 3419171337633496704

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

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

Links

• 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.

Crossrefs

Essentially the same as A001413.

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.

Programs

Python

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

Extensions

a(8)-a(10) copied from A001413 by Andrey Zabolotskiy, May 29 2018

a(11)-a(12) copied from A001413 by Pontus von Brömssen, Feb 28 2024

a(13)-a(16) (using A001413) from Alois P. Heinz, Feb 28 2024

Name edited and "self-avoiding" added by M. F. Hasler, Jun 17 2025