A010570 Number of 2n-step self-avoiding closed paths on the 6-dimensional cubic lattice.

12, 120, 4200, 216720, 13594320, 959431200, 73286046960, 5928739001280, 501123204523440, 43851618007523760, 3946829550070653840, 363607619806646296800

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

The initial term a(1) = 12 corresponds to a cycle which is one step in some direction and one step back. Sometimes such paths, in which an edge is used twice, are not counted as self-avoiding path. - M. F. Hasler, Jun 18 2025

Links

• 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 A8 "Enumeration results for d = 6", 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.

Crossrefs

Cf. A010567 (d=3) - A010569 (d=5).

Programs

Python

def A010570(n): # For illustration - becomes slow for n > 4
    if not hasattr(A:=A010570, 'r'):
       A.terms = [12]; A.weights = 12, 120; I = (0,)*6, (1,)+(0,)*5
       A.paths = (*I,(2,)+(0,)*5), (*I,(1,1)+(0,)*4); A.r = tuple(range(6))
    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

Extensions

a(6)-a(7) from Sean A. Irvine, Jun 01 2018

a(8) from Sean A. Irvine, Aug 17 2020

"Self-avoiding" added in definition by M. F. Hasler, Jun 18 2025

a(9)-a(12) from Clisby et al.'s data added by Andrei Zabolotskii, Jun 25 2025